Flows in Transportation Networks (Mathematics in Science & Engineering, Vol. 90)

(13)

The assumption that total link costs are strictly convex yields the strict inequality ((8), Appendix C) for every f;.’ > fi . (14) Since the feasible reassignment of (5) ensures that at least one link in chain s has a strict increase in flow, it follows that ci(fi’) > ci(fi)

Substituting (15) into (13) gives the strict inequality

C!k)(h) < Cik)(h’) (16) which satisfies the definition (1 1) of a user-optimized flow pattern. To show that the condition is necessary we assume that the condition is not satisfied so that there exist two chains r and s with chain costs C!k)(h) > Cik)(h)

and

hik) > 0 .

(17)

We now deduce that the traffic pattern is not user-optimized.The assumption that total link costs are strictly convex implies that the average chain costs are continuous functions of the flow. (See Appendix C.) Hence, we can always pick a sufficiently small feasible reassignment of

91

16. Flow Depenaent Costs

flow from chain r to chain s (in the sense of ( 5 ) ) to obtain the inequality C:’)(h) > Csk)(h’)> Cik)(h). (18) But this last inequality contradicts the definition ( I 1) of a user-optimized flow pattern. This completes the proof of the theorem. ( c ) Minimum Network Cost Flow Patterns In this section we make use of two additional costs: the marginal cost of flow in a link and the marginal cost of flow in a chain. By marginal cost we mean the change in cost due to a small change in flow, or more precisely, the derivative of total cost with respect to flow. The notation we use for marginal costs in the ith link and jth chain is

and

d di (fi) = - Cfi ci(A)l dfi

Dy’(h) =

(19)

di(fi)a$’. i

Again, the interpretation of the sum in (20) is over all links in thejth chain connecting O(“)to D(k).Throughout this section we assume as before thatfici(f;) is strictly convex and increasing so that the marginal costs defined by (19) and (20) are positive, continuous, and increasing. Since the flows on links in chain j generally depend on other chain flows, the marginal chain cost in (20) is again written explicitly as a function of the flow pattern, h, on the entire network. The total network cost, C = C(h) = Cfi cj(JJ (21) i

sums the total cost of flow in each link over all links in the network. In this section we are interested in deriving a necessary and sufficient condition for a system-optimized flow pattern which minimizes C in (21) subject to the conservation and nonnegativity of chain flows in (1) and (2). THEOREM : A flow pattern h is a system-optimized pattern if and only iffor every commodity k there exists an ordering I , 2, ...,p , p + I , ...,m ( k ) of the chains from O(&)to D@)such that: D‘ik’(h) = D$”(h) = ... Dr’(h)

< Or!,

(h)

< ... Dg:kk)(h)

hik’ > 0; hi” > 0; .-.h(pk)> 0; h r i , = O...h:/k, = 0 .

(22)

92

111. Extremal Principles and Traffic Assignment

This condition implies that it is possible to obtain a subset of chains with positive flow (possibly only one chain) having equal values of marginal chain cost, and chains carrying zero flow have marginal costs greater than or equal to chains with positive flow. The inequalities in (22) can be obtained directly from the theory of nonlinear programming and are analogous to the complementary slackness conditions used in Sect. 15. The proof of the theorem which we shall now give has such special structure that it gives considerable insight into the traffic assignment process. We first define a feasible reassignment of the chain flows by h=h+620

(23)

where S is a vector whose elements Sp) correspond to the elements A$“) of h. While S y ) is unrestricted in sign the flow values from O(k)to D(k) must remain unchanged at

c i

hyk’

=

c

= g(k).

i

(24)

Equation (24) implies that for each commodity we must have

c sp) i

=0

while (23) requires that

Sp)

-h p )

(25)

.

We now proceed to prove that the condition stated in the theorem is suficient. From (22), (25), and (26) it follows that

and that Sik) 2 O

for j

=p

+ 1, ...,m ( k ) .

(28)

Iff’ is the new link flow pattern corresponding to the new chain flow pattern h’ given by (23), the change in network cost is

For a strictly convex functionh ci(jJ,we have shown (Appendix C) that iffi’Z-6, i.e.,f;.’ > orb' < -6, then

fi’ci (-6’) -.A ci (.A)

> (A‘ -A)

4 (.A) .

(30)

93

16. Flow Dependent Costs

Thus, in general, C(h') - C(h) 2

(h'-fi) di(fi)

(31)

*

i

Note that a strict inequality holds when one or morefi'#fi. Because h'# h may lead to cases where fi' ~ f for i all i we must include the possibility that the right-hand side of (31) is zero and C(h') = C(h). Since

A' -fi C(h) - C(h) 2

=

i

k

(32)

a!;'sy',

C c C d i ( f i ) G y ) ~ $=) CC i j k

i

Dy'(h)

(33)

k

using (20). Typical terms on the right-hand side of (33) can be written as

In the first summation, all D$")(h)are equal to Dlk)(h) by (22), and hence, using (27),

so that (34) becomes

By (22) and (28) both factors in each term of the summation on the right are nonnegative so that

C G y ) Dy)(h) 2 0 . i

(37)

Substitution in (33) finally gives C(h') - C(h) 2 0 .

(38)

If the condition of the theorem is satisfied any feasible reassignment of the link flow pattern increases the network cost; any feasible reassignment of the chain flow pattern may or may not increase the network cost. It is rather important to again point out the effects of the nonuniqueness of chain flows in (2) upon the total cost function C(h) in (4) and (21). (See also Sects. 8(c) and (d), Chap. 11.) The analysis we have just provided depends upon the convexity of total link costs in terms of link

94

111. Extremal Principles and Traffic Assignment

flows but the network cost on the left-hand side of (21) is explicitly written in terms of h, not f. There is usually more than one h satisfying (2) when f is fixed; thus even though the chain flow pattern changes, the link flow pattern may not change. If the link flow pattern changes then the strict inequality in (31) and (38) holds. To show that the condition of the theorem is necessary for the minimization of the total network cost, we proceed as in the proof of the previous theorem by assuming that the condition is nor satisfied so that there exist two chains for which the marginal cost of one chain carrying positive flow is strictly greater than some other chain which may or may not carry flow. Mathematically, for some k , r , s , we have

D;“)(h) > D$k)(h)

> 0.

for

(39)

We now deduce that the traffic pattern h is nor system-optimized. We prove this by considering a feasible reassignment A > 0 of flow (in the sense of ( 5 ) ) from chain r to chain s which results in a change of total network cost equal to

+1 C(fi-A)ci(fi-A) i

-.AciU>Ia!:),

(40)

where the summation on the index i applies only to those links not common to both chains. As we indicated earlier (Fig. 16.1) the change in flows on such links is equal to the reassignment in chain flows. What we now want to show is that we can pick A sufficiently small so that the difference in network cost given by (40) is strictly negative. From (11) in Appendix C we obtain the inequalities

(.L+A)ci(J;+A) - f i c i ( J ; ) < di(.&+A)A,

A > 0,

(41)

( f i - A ) ~ i ( f i - A )- f i ~ i ( f J < - d i ( , / i - A ) A ,

A > 0,

(42)

which state that the difference in total link cost due to the reassignment of flow is strictly less than the new marginal cost of link flow times the amount of flow reallocated. Summing terms in (41) and (42) over all links on chains r and s in (40) that are not common to both chains yields the inequality

C(h’) - C(h) < C [di(fi+A)&’ - d i ( f i - A ) & ) ] A i

= [Dtk’(h)- D!“’(h’)]

A.

(43)

95

16. Flow Dependent Costs

But if the inequality of (39) holds for the flow pattern h we can always pick a A sufficiently small and hence a new flow pattern h‘sufficiently close to h so that

Dr)(h) < D!k)(h’), (44) since the marginal cost functions are nondecreasing. Since the term in square brackets in (43) is strictly negative, C(h) - C(h) < 0,

(45)

so that the traffic pattern h does not minimize the network cost. This completes the proof of the theorem. ( d ) Associated Trafic Assignment Problems Surprising as it may seem, both theorems have strikingly similar mathematical structure. The condition of (12) is identical to that of (22) if one replaces Cy)(h) by Dy)(h) and preserves the ordering of inequalities on costs and chain flows. The only substantive difference appears to be that we have substituted marginal costs of chain flow for average costs of a chain. One other observation that may be appropriate at this point is the surprising feature that while one theorem has resulted from strictly local conditions, i.e., comparisons of neighboring routes, the second has been the result of seeking traffic flow patterns which achieve global conditions, i.e., a minimum value for a scalar cost function defined on the entire network. The two theorems enable us to show an intimate relationship between user-optimized and system-optimized traffic patterns. We show that for any given transportation network with given link costs, ci(L),there is an associated problem with related link costs, Zi(si), such that the systemoptimized pattern for the associated problem is a user-optimized pattern for the given network. In fact, we only have to define

It follows immediately that the marginal link costs for the associated problem are

96

111.

Extremal Principles and Traffic Assignment

and hence the marginal chain costs are

By’(h)

=

Cy)(h).

(48)

By the second theorem, a necessary condition for h to be a systemoptimized pattern for the associated problem is given by (22) with B instead of D throughout. By (48), this condition becomes precisely that of (12) which is a suficient condition for h to be a user-optimized pattern of the given network. The consequences of this result for these associated assignment problems are important. We see that we can determine a user-optimized pattern for a given network by solving a minimum cost pattern for an associated problem on the same network. ( e ) A Numerical Example with Four Commodities An example (see [ 171) of the user-optimized and system-optimized flow patterns which are obtained for networks with flow dependent link costs is illustrated in the six-node, ten-link network of Fig. 16.2 having the 0-D trip table: Destinations 1

2

3

Figure 16.2. Network with three centroids, three intermediate nodes and ten links. Link numbers are given alongside the links. Link costs of links 1, ..., 5 are flow dependent and link costs of links 6, ..., 10 are zero.

97

16. Flow Dependent costs

The link costs are given as 1 5-4’

Ci(fi)

=-

CiV;)

=

0 G f i < 5, i = 1, ..., 5 i = 6, ..., 10

0,

(50)

We number the O-D pairs as follows:

O(l)-D(’) : trips from node 1 to node 2, O(’)-D(’) : trips from node 1 to node 3, O‘3’-D‘3’ : trips from node 3 to node 1, O(4)-D(4) : trips from node 3 to node 2, so that our notation for 0-D travel demand becomes g ( l ) = 3. g ( 2 ) = 6 . g(3) = 2 ; g(4) = 5 . 9

(51)

The seven chains that correspond to these four commodities are obtained by listing the distinct links within each set: ((7917 299); (7,3,9)1

with flows h\’),h$*) with flows hi2)

(52) (53)

{(8,4,6)) {(8,4,1,10); (8,5,10)}

with flow hi3) with flows hi4’, hi4)

(54)

M(1) = {(7,1,10); (7,3,5,10)} = M(3)=

M(4) =

(55)

It is interesting to note that link 1 is common to three chains, links 3,4, and 5 each common to two distinct chains, and link 2 to only one chain. The chain flows must satisfy the conservation equations h‘,” + h y ) = 3,

+ h‘;‘) = 5. We see from (52)-(55) that the chain to link flow equations for links 1, ..., 5 can be written as h‘,”

+ h y ’ + h\4’

hi2)

hi’) + hy’

hi3) + h\4)

h y ) + h$4)

= fl, =f2, =f3,

=f4, =f5.

(57)

98

111. Exhmal Principles and Traffic A s s i i e n t

First we obtain the user-optimized flow pattern. To do this we consider the associated problem with link costs calculated from (46) to be

zi(fi> = 0,

i = 6 , ..., 10.

(59)

The total network cost for the associated problem is therefore

c

subject to To obtain the user-optimized flow pattern we minimize (56) and (57). The required solution may be obtained in this simple case by straightforward differentiation or more generally by a nonlinear programming technique but the method is of no concern here. The user-optimized flow pattern is given in Table 16.1. The optimal network TABLE 16.1 THEUSER-OPTIMIZED FLOW PATTERN Optimal chain flows ~-

Chain costs - C:” = 2.794 C;’) = 6.232 C:” = 3.066

Optimal link flows

Link costs

~

h:” = 3.000 hy’ = 0 h‘,2’ = 1.326

h\3’

= 2.000

Ci2) = 3.066 Ci3)= 0.373

h:4’

= 0.316

Cj4’ = 3.166

hi4’ = 4.684

Ci4) = 3.166

hiZ’ = 4,674

f1 =

4.642 = 1.326

~1

f2

~2

= 2.793 = 0.272

f3

= 4,674

~3

= 3.067

f4

= 2.316

~4

= 0.373

f5

= 4.684

c5 = 3.165

cost is C = 43.351. It can be verified that the conditions of the theorem proved in Sect. 16(b) are satisfied. For example,

Ci’) < C$’)

with h\’) > 0, h$”

0

(61)

C 1( 2 )= Ci’)

with hi’) > 0, hi2) > 0.

(62)

=

and

99

16. Flow Dependent Costs

Secondly we obtain the system-optimized flow pattern by minimizing the network cost, given by 5

C = C -A 1 5-f;:’ subject to (56) and (57). The system-optimized flow pattern is given in Table 16.2. The optimal network cost is C=43.238. Again we can TABLE 16.2 THESYSTEM-OPTIMIZED FLOW PATTERN

Optimal chain flows

---

___-__

h y ) = 3.000 h‘,” = 0

hi” = 1.333 hi2’ = 4.667 hi3) = 2.000 h\4’ = 0.332 hi4’ = 4.668

Marginal chain costs

Optimal link flows

___ fi = 4.665 f z = 1.333 fj = 4.667 f4 = 2.332 f s = 4.668

~

D\” = 44.643 D‘,” = 90.361 D:” = 45.015 DS2)= 45.015 DY’ = 0.702 0‘:’= 45.346 Di4) = 45.346

Link costs -

CI

= 2.988

cz = 0.273 ~j

= 3.001

c4 = 0.375 cg

= 3.012

verify the conditions of the theorem proved in Sect. 16(c). The marginal link costs are 5 i = 1, ..., 5 div;) = (64) ( 5 -A)2

’

and from the table we see that the marginal chain costs satisfy, for example, Dl’) < DS1)

with h\’) > 0, h y )

0

(65)

D (1 2 )= 0”)

with hi2) > 0, hi2) > 0.

(66)

=

and To see just how complicated the interactions between chain flows and chain costs really are, consider the effect of a small decrease in the chain flow hi4). hS4) must be increased to preserve feasibility. This reassignment decreases fl and f4 and increases f5 because these links are incident upon both chains 1 and 2 of commodity 4. Obviously c1 and c4 must decrease while c5 increases. These changes then force C!’),C\”, C!”, and

100

111. Extremal Principles and Traffic Assignment

CI4) to decrease while Cl') and Ci4) must increase. In other words, the reassignment of one chain flow affects the costs of six out of the seven chains that we have considered thus far! ( j ) Congested Assignment

Despite the elegance of the theoretical analysis of extremal principles for flow-dependent costs, it has not led to assignment procedures or programs which can handle large transportation networks. Many congested assignment programs, as they are called, have been used in transportation planning but the present state of the art is such that no one program has been widely accepted. Most congested assignment procedures are iterative. For certain link costs, traffic is assigned and then the link costs recalculated, traffic reassigned, and so on. It is a pious hope that this iterative procedure should converge, and although possible oscillations can be overcome or damped out the resultant assignments become less meaningful. For the special case when the link costs are step functions, it is possible to formulate the congested assignment as a linear program which can cope with reasonably large networks. For example, Charnes and Cooper [24] have formulated a multicopy traffic assignment model which, in the link flow notation of Sect. 8(d), Chap. 11, can be expressed as the following mathematical program :

Ef" = g", f =cfa

(68)

< u,

a

c(f)Tf = C(min),

(70)

where fa is the unknown I x 1 link flow vector with elements equal to the link flow of copy a ; g" is the given n x 1 copy flow vector; E is the n x I node-link incidence matrix; u the n x 1 vector of link capacities; and c(f) the I x 1 vector of link costs. The increase in link cost with flow is represented by a step function with a separate parallel capacitated link for each step., For example, two parallel links i = 1,2, connecting the same nodes, one with link cost cI and capacity u1and the other with link

101

16. Flow Dependent Costs

cost c2 and capacity u 2 , and c2 > c,, give the piece-wise linear function

Thus, by coding the network with suitably chosen parallel links where necessary, the multicopy assignment is still expressed as a linear program and can be solved by the usual LP algorithms. Charnes and Cooper illustrate the theory with a multicopy assignment to a street network with 18 nodes, 27 links, and 1 1 origins giving 1 1 copies. A more detailed example has been given by Pinnell and Satterly [25]. Tomlin [26] has explicitly formulated congested assignment as a minimum network cost multicommodity flow problem. With the notation of Sect. 8(d), Chap. 11, the problem can be expressed as the following mathematical program : hy) 2 0,

1 h p ) = g(“), i

(72) (73)

where Cjk)istheroutecostfor thechain m y ) .Ifthe costsare step functions, this program is a linear program and can be solved by the multicommodity flow algorithms. Unfortunately these are inefficient, and the complete enumeration of the 0-D chains for a large network would be impossible. This may be overcome by a column generating technique which introduces new chains calculated by a cheapest route algorithm. Use is made of the dual linear program: v(~)

unrestricted in sign,

Pi 2 0, V ( k ) - 1Pi < cp, (i)

1v(k)g(k)- 1piui = V(max).

(76) (77) (78) (79)

In (78) the dual variables pi are, by virtue of (74), independent of the commodity k ; the summation in this equation follows the convention adopted in (43), Sect. 15. Various techniques have been tried in an attempt to improve the efficiency [27], but practical applications have in the main been restricted to small networks.

102

111. Extremal Principles and Traffic Assignment

17. Notes and References

Wardrop, J. G., Some Theoretical Aspects of Road Traffic Research, Proc. Znst. Civil Eng., Part 11, 325-378 (1952). This interesting paper and the ensuing printed discussion summarizes some of the early research work done at the Road Research Laboratory, England, which has pioneered and stimulated the dramatic recent growth of traffic research throughout the world. The paper emphasizes the value of a theoretical approach to traffic problems and illustrates this with an analysis of the following: flow and speed of traffic, frequency of overtaking, capacity of road systems, signal-controlled intersections, formation of queues, distribution of traffic over alternative routes, before-and-after studies. Although the paper was published some years ago, it still serves as an excellent introduction to traffic theory. It is curious that as often as not the first extremal principle enunciated by Wardrop is referred to in the literature as his second, and the second as his first! [I]

Murchland, J. D., Bibliography of the Shortest Route Problem, London Business Studies Report, LBS-TNT-6.2 (1 969). The reports from the short-lived Transport Network Theory Unit of the London School of Economics and Political Science (and subsequently of the London Graduate School of Business Studies) form a very significant and lively contribution to transportation science, and it is regretable that few of the reports have found their way into wellread journals. Report No. 6.2 is a second revision of an extensive bibliography, containing 97 references. Murchland classifies the shortest route algorithms into three main categories-tree building, matrix, and partitioned. [2]

Dreyfus, S. E., An Appraisal of Some Shortest-path Algorithms, Operations Res., 17,395-41 2 ( 1 969). The purpose and scope of this interesting paper can be best summarized by quoting first from the author’s introduction and then from his conclusion: “... it is hoped that our somewhat skeptical survey of current literature will put the interested reader on guard and perhaps save him, or his digital computer, considerable time and trouble;” and “Our interest has not been definitive solution, but rather to clear the air by presenting both some important methods and references and some critical comments and warnings.” [4] Kirby, R. F. and Potts, R. B., The Minimum Route Problem [3]

17. Notes and References

103

for Networks with Turn Penalties and Prohibitions, Transportation Res., 3, 397-408 (1969). This paper surveys the literature and resurrects an important paper which had been apparently overlooked. It also points out an erroneous procedure which has been suggested in the literature. Bellman, R., On a Routing Problem, Quart. Appl. Math., XVI, 87-90 (1958). This comparatively early paper is important because it is one of the few in the literature which gives a precise statement and proof of a minimum route algorithm. The present text uses the dynamic programming functional equation technique described in this paper. An earlier and similar analysis of the shortest route problem is found on pp. 52-54 of the monograph by Beckmann [6], Sect. 12, Chap. 11. [5]

Dijkstra, E. W., A Note on Two Problems in Connexion with Graphs, Numer. Math., 1, 269-271 (1959). This short paper briefly describes the algorithm which is analyzed in detail in the text. We have purposely avoided attaching authors’ names to algorithms because of the general confusion in the literature on cheapest routes. [6]

Caldwell, T., On Finding Minimum Routes in a Network with Turn Penalties, Comm. A C M , 4, 107-108 (1961). This important paper seems to have been overlooked and has, for example, escaped inclusion in Murchland’s 1967 bibliography. Perhaps one reason for its neglect is that it does not formulate a practical algorithm directly applicable to networks as they are usually coded in transportation planning. [7]

Brokke, G. E., Urban Transportation Planning Computer System, American Association State Highway Oficials Conference, Minnesota (1967). The Bureau of Public Roads has cooperated with State Highways Departments for many years in establishing transportation planning surveys. Their computer packages developed in the late 1950s for use on first generation computers such as the IBM704 have been widely used and have been largely responsible for defining the analytic computer oriented approach to the planning process. Early in the 1960s, the computer packages were revised for the second-generation computers (e.g., IBM 7094) and the Traffic Assignment Manual (1964) is one of the excellent publications which explain in detail the use of the programs. The Bureau is currently rewriting the programs for the [S]

1 04

111. Extremal Principles and Traffic Assignment

third generation computers (IBM 360), and the new tree or vine building programs such as described in this reference include an option which permits turn penalties and prohibitions to be correctly counted. Control Data Corporation Data Centers Division, Users’ Manual, Transportation Planning System for the Control Data 3600 Computer (1965). Control Data Corporation has developed a transportation planning system, called TRANPLAN, which has been widely used as an alternative to the BPR program packages, although basically the computer programs are very similar. The system is being rewritten for the 6000 series of Control Data computers. [9]

[lo] Sema, Group Metra France, Le Modele ATCODE (1964). Metra has provided a significant contribution to transportation planning by its development of a comprehensive package of computer programs, including generation, distribution, modal split, and assignment.

Kirby, R. F., A Minimum Path Algorithm for a Road Network with Turn Penalties, Proceedings of the Third Australian Road Research Board Conjerence, Part Z, 4341142 (1966). A fully documented version of this algorithm has been lodged with the National Association of Australian State Road Authorities and has been used with success in several metropolitan transportation studies. The program has been written for a 32K Control Data 3600 Computer for networks of up to 3000 nodes (including a maximum of 650 centroids) and 12,000 links with 32 turn penalty types available for each turn. The program has been modified for use on a Control Data 6400 Computer and includes a facility for specifying intersection types as well as turn types. [I I]

[ 121 Wachs, M., Relationships Between Drivers’ Attitudes Toward Alternate Routes and Driver and Route Characteristics, H.R. B. Record, 197, 70-87 (1967). This paper reports a home-interview study designed to analyze the factors which drivers consider important in the choice of routes for different purposes-work trip, shopping, and a trip to visit a friend. The drivers expressed their preferences for access controlled routes, shortest routes, safety, congestion, strain, pleasant scenery, etc. A careful statistical analysis led to the conclusion that reasonably strong relationships do exist between the attitudes of drivers toward the type of

17. Notes and References

I05

route they seek when they make a trip, and the characteristics of the drivers, their trips, and the routes to which they have been exposed. [I31 Jansen, G. R., A Pilot Study in Trip Assignment, I.T.T.E. Graduate Report, University of California, Berkeley (1966). This report gives the result of a survey made of drivers’ choices of routes between the Plaza of the Golden Gate Bridge and the downtown office of the California State Automobile Association. The travel times on roads of the network were measured by the floating car method, and the results differed quite markedly from those incorporated in the BATS networks. From travel diaries kept by 17 drivers, Jansen found that five significantly different routes were chosen and of these the most popular (in the morning peak) was the next-to-longest in travel time! Turn penalties and prohibitions were not taken into account. Stover, V. G., The Texas Large Systems Traffic Assignment Package, Trafic Quart., 21, 339-354 (1967). This paper describes a battery of programs TEXAS-BIGSYS which can carry out cheapest route traffic assignments for a main road network with 16,000 nodes (including 4800 centroids) and 64,000 links. This may be a significant computing achievement but hardly a useful tool for the planner. [I41

Michaels, R. M., Attitudes of Drivers Determine Choice between Alternate Highways, Public Roads, 33,225-236 (1965). This paper reports a study carried out on driver’s choice between two routes-one the Maine Turnpike from Kittering to South Portland, U.S.A., and the other the parallel rural highway US 1. Over three thousand drivers were questioned, and from their answers the author concluded that drivers did have stable attitudes which correlated with their choices. Test drivers were used to measure the tension or stress characteristics of the alternate routes and these were shown to be a significant factor in determining the diversion to the freeway route. [IS]

[I61 Burrell, J. E., Multiple Route Assignment and its Application to Capacity Restraint, Fourth International Symposium on Theory of Trafic Flow, Karlsruhe (1968). This interesting paper reports on a novel assignment program developed in conjunction with the London Transport Study, England. The rectangular link cost probability function is characterized by the mean absolute deviation, its quotient with the mean link cost being taken the same for all links. The possibility of choosing one of eight link

106

111. Extremal Principles and Traffic Assignment

costs still gives a unique 0-D route, but within the network many links will be used thus spreading out the traffic flows.

[I71 Jorgenson, N. O., Some Aspects of the Urban Traffic Assignment Problem, 1.T.T.E. Graduate Report, University of California, Berkeley (1963). It is unfortunate that some of the material in this report did not appear earlier in the open literature. Partly because of this, we have given Jorgenson’s treatment of the link flow linear programming approach to the two extremal principles in detail, although his statement of the extended first principle has been somewhat modified. His derivation of the equivalent mathematical problems for user-optimized and minimum network cost traffic patterns with convex link costs predates other published formulations of the model. In the context of transportation problems, he showed that the flow versus travel time relationships were identical in structure to the Kuhn-Tucker optimality conditions of an equivalent extremal problem. While the author is primarily concerned with a link flow formulation, we have chosen to emphasize the chain flow aspects of traffic assignment. [ 181 Dantzig, G . B., Linear Programming and Extensions, Princeton Univ. Press, Princeton, New Jersey (1963). Of the many texts on linear programming, we have chosen to refer to this one because it is authoritative, readable, and contains chapters on network flows and the cheapest route and minimum network cost problems. In Chap. 17, the author emphasizes the importance of the relationship between trees and basic solutions. In a lucid paper (SIAM Rev., 10,371-372 (1968)), A. F. Veinott and G . B. Dantziggive aconcise proof that the total unimodularity of a matrix A is a necessary and sufficient condition for integral inverses of every basis of the system of inequalities x > 0, Ax < b, where A and b are integral. The node-link incidence matrix E in (4) (Sect. I5), the augmented matrix (E, I) implicit i n (20) and (21) (Sect. 15), the link-chain incidence matrix A implicit in (39) (Sect. 15) are examples of totally unimodular matrices. Thus, the linear programs of Section 15(b) yield integral basic solutions for link and chain flows when the given flow values and link capacities are integers. The property of total unimodularity does not extend to the class of augmented incidence matrices that characterize multicommodity and multicopy flows.

[I91

SHARE Program SDA 3536,Out-of-Kilter Network Routine, OKF3 (1967).

17. Notes and References

107

This program solves the minimum network cost single commodity problem for a capacitated network with given constant link costs. The abstract for this well-documented program is as follows: “An independent routine to solve capacitated network flow problems using a method in which a measure of optimality is not worsened on any iteration. Flows have upper and lower bounds which may be positive or negative. No initial feasible solution is needed. Has provision for solving problems which vary slightly from previously solved problems in minimal machine time. Source language is Fortran IV.” This program and variants of it have been widely used. It is interesting to note that our simplified presentation of the out-ofkilter algorithm, which assumes an initial feasible solution, must have been similar to that first drafted by Fulkerson. In his original paperAn Out-of-Kilter Method for Minimal-cost Flow Problems, SIAM J., 9, 18-27 (1961), which is the basis for the relevant section in the Ford and Fulkerson text ([l], Sect. 11, Chap. 11)-Fulkerson expresses his appreciation to Dantzig “whose criticism of an earlier version of this paper in which the initial x was assumed feasible, led us to reconsider the problem from the standpoint of infeasible x.” Nash, J., Non-Cooperative Games, Ann. of Math., 54, No. 2, (September 1951). This highly original paper develops the notion of an equilibrium point for n-person games in which players do not cooperate or form coalitions. The author shows that every such game has at least one equilibrium point. The analogy with traffic assignment on a transportation network is that each commodity represents a player; a traffic flow equilibrium corresponds to the case where travelers having routes with a given 0-D pair minimize their travel costs when the routes used by all other commodities are held fixed. The traffic patterns of ( I 1) (Sect. 16) are more properly referred to as Nash equilibria. Possibly one of the most important requirements that is implicitly required in Nash’s theory (but not necessarily attained in real-life transportation networks) is the availability of complete information on the status of alternate routes. [20]

[21]

Dafermos, Stella C., and Sparrow, F. T., The Traffic Assignment Problem for a General Network, NBS J . Res. Ser. B, 73, 91-118 (1969); Dafermos, Stella C., An Extended Traffic Assignment Model with Applications to Two-way Traffic, Transportation Sci.,5, 366-389 (1971).

I08

111. Extremal Principles and Traffic Assignment

These two papers (the later one more readable than the first) give a rigorous mathematical treatment of traffic assignment problems for flow dependent costs. The careful distinction made between user-optimized and system-optimized traffic patterns has been followed in our text, together with the statements and proofs of the two important theorems we give in Section 16. The authors also develop an interesting algorithm which begins with an initial feasible flow pattern and by means of an “equilibration operator” constructs a sequence of feasible flow patterns which converges to the user-optimized solution. Because of the relationship between the user-optimized and system-optimized solutions as discussed in Sect. 16(d), this algorithm also provides a method for solving the minimum network cost problem. Beckmann, M. J., On the Theory of Traffic Flow in Networks, Trafic Quarf.,21, 109-1 17 (1967). The author formulates an abstract and general model of flows in transportation networks. Most of the important contributions in this field prior to 1967 can be viewed as special cases of his model. Although his notation differs substantially from our own, our material covers the five classes of relations that specify his transportation network model: (1) average costs as a function of link flow, (2) the relationships between 0-D travel demand and the cost (or time) of travel, (3) the relationship between route costs and link costs, (4) the conservation equations which state that link flows are the sum over all commodities of commodity flows on each link, and (5) Kirchhoff’s conservation equations for each commodity or copy. He also recognizes the existence of a “suitable” mathematical function whose extremal values allow one to characterize the equilibrium flow patterns in transportation networks. He does not, however, explicitly formulate them in terms of Nash equilibria. See also [ 6 ] , Sect. 12, Chap. 11. [22]

Kitchen, J . W., Calculus of One Variable, Addison-Wesley, Reading, Massachusetts (1968). Chapter 6 of this book has an excellent and simple two-dimensional treatment of convex functions and convex sets. The mean-value theorems, the three-chords lemma, and some of the inequalities that we use are stated and proved in this very readable text. Although the material can be used as background for an advanced course, this calculus book is often used at the freshman and sophomore college level. [23]

[24]

Charnes, A. and Cooper, W. W., Multicopy Traffic Network Models, in Theory of Trafic Flow (R. Herman, Editor), 84-96, Elsevier, Amsterdam (1961).

18. Problems

109

This paper was presented at the. First International Symposium on the Theory of Traffic Flow held at G. M. Research Laboratories, Michigan, U.S.A. The theory is illustrated with an interesting example computed from a modification of data pertaining to a small town in Indiana. Pinnell, C. and Satterly, G. T., Analytical Methods in Transportation : Systems Analysis for Arterial Street Operations, J. Engrg. Mech.-Diu., Proc. Amer. SOC.Ciu. Eng., 89, 67-95 (1963). The title of this paper perhaps disguises the content, which is a detailed example of a multicopy assignment to a network with 25 nodes, 176 links, 12 origins, and 3 destinations. The copy flows were identified as those with particular destinations, giving a total of 3 copies. All except 4 links were taken as uncapacitated, and on these four a step cost function was assumed. The same example was solved using a discrete version of Pontryagin’s principle by Funk, M. L., Snell, R. R., and Blackburn, J. B., (see J. Hway.-Div., Proc. Amer. SOC.Civ. Eng., 93,95-113 (1967)). [25]

[26]

Tomlin, J. A., Minimum Cost Multicommodity Network Flows, Operations Res., 14, 45-51 (1966). In this paper, the author gives a node-link and a link-chain formulation of the minimum network cost problem and discusses their equivalence. A more detailed discussion is contained in the author’s PhD thesis : Mathematical Programming Models for Traffic Network Problems, University of Adelaide, South Australia, 1967. [27]

Gibert, A., A Method for the Traffic Assignment Problem, Report LBS-TNT-95, London Graduate School of Business Studies (1968). This paper suggests a method for improving the efficiency of a multicommodity assignment program by storing and not discarding cheapest routes determined for nonoptimal solutions. Experience shows that for each commodity flow between a particular O-D pair, few of the possible chains are used, and these are usually determined in the first few interations.

18. Problems 1 . For the undirected network illustrated in Fig. 18.1, use the algorithm described in Sect. 14(b) to find the cheapest route tree with node 1 as

I10

111. Extremal Principles and Traffic Assignment

home node. Is the cheapest route from node 1 to node 6 altered if the link costs are all increased by 2 units?

Figure 18.1. Undirected network with link costs as indicated.

2. Find a second-to-cheapest path from node 1 to node 6 in the network illustrated in Fig. 18.1. Describe a general algorithm for finding secondcheapest routes on a network.

3. Consider an undirected connected network with given link lengths. Is the following statement true: if a dearest path from node n, to node n, passes through node ni, then that portion of the path from n, to ni is a dearest path from n, to ni? Illustrate your answer by considering the network in Fig. 18.1 and taking n, = 1, ni = 2, n, = 3. 4. For the network [ N ; L] illustrated in Fig. 18.1 define cii = 0, i E N, and c i j = co, ( i ,j ) # L. Compute the following for i,j = 1,2, ..., n : Step k = 0:

C!?) = c.. . IJ 1J 7

Steps k = 1,2, ..., n:

cjj")= min [c!,"I ) , Cg-1 ) + c L5- "1 . Show that C$) = C: is the cost of a cheapest path from node i to nodej, and justify the algorithm.

5. With the same definitions as in Problem 4, compute successively matrices of order k = 1,2, . . . , a with elements Cij")as follows: Step k = 1 : c(1) = 0. 11 9

111

18. Problems

S t e p s k = 2 , 3 ,..., n: For i , j = 1,2,...,k - I , define (a) Ci(kk)= minj[C$-l)+cjk] ; (b) c$)= minj[ckj+C$-’)] ; (c) cg) = 0 ; (d) C$) = min [C$--”, Cl[)+ C $ ] . Show that C$)= C$ is the cost of a cheapest path from node i to nodej, and justify the algorithm. 6. Consider the following statement for a network without turn penalties in relation to (15), Sect. 14: If O < C * ( n l , n j ) - C * ( n , , n i )< c(ni,nj), then there is no cheapest path from nl to nj which passes through ni. Is this statement true or false?

Figure 18.2. Network with prohibited turn. This network is the same as that in Fig. 14.4, except that the links are now numbered 1,2, ... , 7 .

7. For the network in Fig. 18.2, suppose that the link costs are ~ ( l )= ~ ( 2 =) ~ ( 3 )= 20,

~ ( 4 )= ~ ( 5 = ) 60,

4 6 ) = c(7) = 90,

and that the turn penalties are P(1,2)

= P(6,5) =

0,

P ( 1 , 5 ) = P(3,4) = P(4,I)

=

P(593) = 5,

P(1,7) = 15,

with all other values of p = 00. Compute the cheapest route tree with link 6 (or more specifically the beginning of this link) as home link.

112

Extremal Principles and Traffic Assignment

111.

8. In an attempt to account for turn penalties, the tree-building algorithmdescribedinSect. 14(b) is modified as follows: to the condition (nk-

(xk-

1 9

I , xk-

1)

in (i) for steps k = 2,3, ..., n is added “and the sequence of the three nodes P(nk- I), nk- I , nj corresponds to an allowable turn”, etc. Show that this new plausible algorithm is incorrect by applying it to the following network: N = {1,2,3,41,

L = {(l,2), (193)9 (194)Y (2931, (3,411, c(1,3) = 1,

c(1,2) = 3,

~ ( 2 ~= 3 )1,

c(1,4) = 6,

~ ( 3 , 4 )= I ,

turn from link (1,3) to link (3,4) is prohibited. 9. Consider the directed single 0-D capacitated network illustrated in Fig. 15.2 with corresponding link capacities and costs as follows: Link Link number capacity I ui

-

..

1

2 3 4 5

6 7 8 9

10

fi

Dual variable Pi

2

5 1 2

0 0 I

6 8 3 5 1 I 1

3 0 3 0 0 6 0

0

Link cost ci

__

~

6 8 2 4 1 5 3 4

8 9

I 4

Link

flow ___

0 0 0 0 0 0

(i) Show that the tabulated link flowsf, give a feasible network flow with flow value g = 6 and network cost C = 46 units. (ii) Show that the tabulated dual variables pi give a feasible solution of the dual LP with v = 8 and V = 46 units. (iii) Verify the complementary slackness relations (47)-(50) (Sect. 15). (iv) Interpret the optimal chain flow solution in relation to the extended form of the first extremal principle.

113

18. Problems

10. Repeat Problem 9 using the same network and the same link capacities and costs but take

g = 11,

and

fT = [ 6 5 2 4 0 5 2 1831,

v = 10,

and

pT = [lOOOOlOOlO].

Figure 18.3. Multiple 0-D network for traffic assignment. 0, and O2 are the origins, D, and D2are the destinations, and the links are numbered 1,2, ..., 11.

11. Figure 18.3 represents a directed network with 2 origins 0 , , 0 2 , 2 destinations D,, D,, 6 intermediate nodes, and 11 links numbered i = 1,2,..., 11.

(i) Enumerate all chains from the origins to the destinations and determine their route costs if the link costs are given by Link :

i

1 2 3 4 5 6 7 8 9 1 0 1 1

Linkcost: ci 3

1

5 I

1

2 4

I

I

1

3 .

(ii) The O-D flows are given by the trip table

: [ :: ::1 :: D, D,

.50 50

100 = U.

Assuming that the links are uncapacitated, assign these trips and determine the commodity and link flows using all-ornothing cheapest route assignment. What is the network cost? 12. With the same data as in Problem 1 1, determine the cheapest routes

114

111. Extremal Principles and Traffic Assignment

to the destinations from origins 0, and O,, and find the corresponding copy and link flows using all-or-nothing cheapest route assignment.

13. Repeat Problem 12 using copy flows to D, and D,. 14. With the same data as in Problem 1 I, suppose that the link i = 4 has a capacity u4 = 50 units, all other links remaining uncapacitated. Using all-or-nothing cheapest route assignment, assign the 0-D trips to the network (i) in the order 0,-D,, O,-Dz, 0,-D,, O2-D2, (ii) in the order 0,-D,, 02-D1, 0,-D,, 0,-D,. Determine the link flows and network cost in each case. 15. With the same data as in Problem 14, determine the network flow which minimizes the network cost. 16. Links 6 and 7 in the network in Fig. 18.3 have link costs given by (.6(f)

C7(f)

=

=

[ ’+ (:’+

0< f

< 20,

f 2 20,

0.4(f - 20),

O<
f 2 30.

All other links have constant link costs Link :

i

1 2 3 4 5 8 9 1 0 1 1

Link cost: ci 3 3

5 5

1

1

3

1

3

For the trip table as in Problem I I , assign the 0-D trips to give a useroptimized flow pattern. Are the optimal chain flow and link flow patterns unique? What is the network cost? 17. For the same data as in Problem 16, find a system-optimized flow pattern. Are the optimal chain flow and link flow patterns unique? What is the minimum network cost?

18. Consider a single 0-D capacitated network [ N ; I,] with origin node 1 and destination node n. By introducing an uncapacitated return link (n,I), formulate the maximal flow problem as a linear program, both primal and dual, and prove the max-flow min-cut theorem.

CHAPTER

IV TRIP DISTRIBUTION

19. Introduction

Trip distribution plays an important role in the analysis and network evaluation phases of the transportation planning process. The concept of trip or trip interchange is somewhat loosely and ambiguously defined. Usually it refers to interzonal journeys from one zone (origin) to another zone (destination) and to intrazonal journeys within a zone. Sometimes the trips refer to total person trips, sometimes to vehicle trips. The number of trips is estimated for a particular period of time and may refer, for example, to an average weekday or peak hour. The trip numbers, therefore, have the dimensions of flow and the two terms will be used interchangeably. The trips are usually classified according to purpose, the main stratifications being home-based work trips, home-based nonwork trips and nonhome-based trips. This awkward terminology is somewhat self-explanatory. Special consideration may be given to external trips, i.e., trips which have origin or destination outside the study area. The essential problem in trip distribution is to determine, from the estimated number of trips produced at and attracted to each zone, the number of interzonal and intrazonal trips. The numbers of trips can be regarded as elements of a trip table or distribution matrix or as flows on a traffic desire network. In the planning process, trip distributions are evaluated in conjunction with traffic assignment models, such as have been described in the 115

116

IV. Trip Distribution

previous chapter. Most trip distribution models use as parameters the interzonal travel times or costs but in general these depend on the network and the traffic assigned to it. Thus, the output of the traffic assignment program is required as an input to the trip distribution program. On the other hand, the output of the trip distribution program, giving the interzonal trips, forms an essential input to the traffic assignment program. This feedback between traffic assignment and trip distribution is especially important in the evaluation of new networks. We shall in this chapter first formulate trip distribution as a mathematical model and then analyze in some detail a variety of these models. The first we shall discuss is the Hitchcock model [I] which in the literature on network flows is usually associated with the transportation problem. This model features both extremal principles which were discussed in the previous chapter. Next we proceed to analyze a class of distribution models which are based on the concept of network entropy. Included in this class is the gravity model, which has served as one of the mainstays of transportation planning. In the following section we give a brief description of opportunity models, with special attention to a preferencing model. In order to interrelate the concepts of this chapter and the previous one, we conclude with some models which simultaneously consider trip distribution and traffic assignment. 20.

Model Formulation

Formulated as a mathematical problem, trip distribution requires the determination of the number of tripsfii from centroid i to centroidj given the total number of trips a, produced at i , the total number of trips hi attracted t o j , and the cost cij of a trip.from i toj. The essence of the mathematical model is the functional dependence o f L j on the a , hi, and cij. As usual, cost may be interpreted as travel time, distance or a combination of these. The cii are often determined in the transportation planning programs as output of a “skim trees” routine giving the travel times of the shortest (i.e., quickest) interzonal routes. The quantities cii are chosen as mean travel times for intrazonal trips. There are various constraints which may be regarded as desirable properties of a trip distribution model. For example, one might require that theAj satisfy the conservation laws Cfij

i

=

ai,

(1)

20.

117

Model Formulation

with the understanding that

C ~i

=

I

1 bj = U, i

(3)

where u is the total number of trips or flow value. Although this seems an obvious mathematical requirement, some models do not force exact equality in (1) and (2), and there is some justification for this in transportation planning, because the values of ai and bj are usually not known with any great accuracy. In addition to conservation, the model may be required to satisfy the compressibility and separability constraints described in Sect. 8(e), Chap. 11. For example, if centroids n - 1 and n of a traffic desire network with n centroids are combined, compressibility requires the new trip numbersf;.;., productions ai', attractions bj', and total trips u' to be related to the old by the equations f.'. = f.. IJ

X - 1 . j

fi:n-1 fi-l.n-1

= a.I7

0.'

IJ'

b.'

=

b.

J 9

(4)

=fn-I,j+fnj,

a;-l

=

an-l + a n ,

(5)

=A,n-l

bA-1

=

bn-1 + b n ,

(6)

+fin,

=fn-l,n-l

+fn-l.n

+&,,-I

+Ln,

(7)

for i,j = I , 2, ..., n - 2. Note that

v' =

1a.' = C a, =

0.

(8)

Separability requires that if node n were removed, then the new trip numbers, productions, attractions, and total trips should be f.EJ'.= f SJ' ..

a.'

= a.I - f. in 9

bj'

=

U' =

(9)

(10)

bj -hi, u - an - b,,

f o r i , j = 1,2,...,n-1. Both compressibility and separability are desirable properties of a trip distribution model, because they imply some independence of the particular manner in which the study area has been subdivided into zones.

118

IV. Trip Distribution

Finally we may wish to force the model to give nonnegative integral values ofAj. In some models this is automatically satisfied; in others it is an important constraint. In practice it is not possible or even desirable to try to satisfy all these requirements, and in describing the various models which have been proposed, we shall indicate which of the various properties they possess. The distribution model has been formulated using ai,bj. It is sometimes convenient to introduce dimensionless quantities p i j , ui, u j defined by

xi,

so that xui

=

xuj

= &Iii

i

i

=

I.

i.i

These normalized quantities are particularly important for the probabilistic interpretation of trip distribution models. 21.

Hitchcock Model

The classical transportation problem, formally stated by Hitchcock in his 1941 paper as “The Distribution of a Product from Several Sources to Numerous Localities,” can be interpreted as a trip distribution problem in the following way: the traffic desire network is regarded as a bipartite graph in which the m trip origins are numbered i = 1,2, ..., m, and the n trip destinations are numberedj = 1,2, ..., n. If the number of trips originating from i is denoted by ui, the number of trips with destination j by hi, and the total number of trips by u, then m

n

I

I

C ai = C bj = U . The cost associated with a single trip from i t o j is denoted by cij, and this is usually determined, on the basis of the first extremal principle discussed in Chap. 111, as the cost for the cheapest route between i andj. The number of tripshj between i and j is then determined, on the basis of the second extremal principle, by minimizing the total network cost C .

119

21. Hitchcock Model

The Hitchcock trip distribution model is thus formulated as the linear program

i

j

cijfij = C(min),

with i = 1,2, ...,m,a n d j = 1,2, ..., n. The dual of this LP is ai

1aiai+

+ <

Cij’

(6)

ai, B j

unrestricted in sign,

(7)

Bj

bjBj = V(max), (8) where the dual variables ai,Bj are usually called implicit prices. The various algorithms for the solution of this special LP are too well known and documented to bear repetition here, and the interested reader can find excellent and exhaustive discussions in reference [l], Sect. 11, Chap. I1 or [IS], Sect. 17, Chap. 111. Instead, we shall briefly interpret the minimum cost solution as an optimal trip distribution, and describe its properties. An important property of the optimal solution is that, in general, not more than m + n - 1 of the mn optimal trip numbersf; can be nonzero. In other words, the optimal trip distribution represents a concentration of the trips on relatively few of the possible 0 - D desire lines. This concentration could be quite erroneous for certain trip purposes, such as recreation and nonhome-based trips, which are likely to be rather spread out over an area. On the other hand, this concentration is evident for home-based work trips because of the tendency for people to choose to live near where they work. In the afternoon peak period, for example, we might regard the origins of the work-to-home trips comprising, say, 50 zones and the destinations 300 zones. The optimal trip distribution would then be concentrated on about 350 0-D pairs, which represents an average of 7 possible destination zones for each origin zone. It is clear, then, that the chosen subdivision of the area into zones and the trip purpose under consideration will be an important factor in deciding whether the Hitchcock model would be an appropriate distribution model.

120

IV. Trip Distribution

The Hitchcock model evidently satisfies the conservation requirements and also automatically ensures nonnegative integral trip numbers. The model can be forced to satisfy the compressibility constraint by an appropriate but somewhat artificial choice of interzonal costs. Suppose that the destination zones n- 1 and n are combined. Then compressibility requires that =hi, ai' = a,, b.' = 6.J ' (9)

+

b:- I = b,,- 1 b,, , (10) for i = 1,2, ...,m, and j = 1,2, ..., n- 2. To achieve this, the new costs ~ f , ~ have to be calculated from the optimal solution of the original problem. If h7 denotes the optimal solution of the original primal problem, and mi*, Bj* the optimal solution of the original dual problem, then the new costs are chosen as follows: &n-I

=fi,n-l

+fin,

c!. = c.. EJ iJ' =

bn-1ci,n-l bn-1

either

+ bncin

+ bn

fi:n-l

(1 1) 3

> 0, f; > 0, or

A:,,-,

=

0,

=

0,

(12)

for i = 1,2, ..., m, and j = I , 2, ..., n-2. Equation (12) indicates that if the desire lines between zone i and zones n- 1 and n are either both used or both not used in the original network, then for the compressed network the cost of a trip between zone i and n is a convex linear combination of the original costs, using the attractions b,- and 6, as multipliers. It is evident that this combination would not be satisfactory to describe the case when one and only one of the desire lines is used. For suppose ci,"were very large, forcingf; = 0, but that c i , n - were sufficiently small so that f;:,,-, > 0. The convex linear combination of c ~ , ~ -and , c ~ , ~ could well be large enough to incorrectly force A:*,- = 0. In this particular case, (1 3) becomes

because of the complementary slackness relations.

22.

121

Entropy Models

That this choice of new costs gives the required compressibility of the model is most easily verified by appeal to duality theory. It is claimed that iff:, ai*, Pi* are the optimal values of the primal and dual variables for the original network, then

f! EJ? = f.?EJ ’ x:-1 =.h:n-t

.f* = ai*,

(15)

+f:, B!*

(16)

P.*J ’

=

(17)

for i = 1,2, ..., m, and j = I , 2, ..., n - 2, give the optimal values of the primal and dual variables for the compressed network. It is not difficult to check that these values give primal and dual feasible solutions and that the complementary slackness relations and

> 0 => .:*

+

= c&-,

,

(19)

.;* + p;*- 1 < c;,.- 1 =>fi,*n-= 0,

(20) are valid, giving optimality. It is an immediate consequence of (1 1) that since C* = V* and C‘*= V’*,

C‘*

=

V’* = 1ai’.f* + m I

n- 1

1 bit&*, I

C aiui* + 1b j p j * , I I = v* = c*, m

n

=

i.e., the optimal network cost is unchanged. Thus, with appropriate choice of interzonal costs, the Hitchcock model can be made to satisfy the compressibility requirements. However, it does not possess the separability property. 22.

Entropy Models

(a) Network Entropy

The concept of entropy, familiar in thermodynamics and information theory, provides a useful unifying basis for a class of distribution models, of which the gravity model has been one of the most commonly used in

122

IV. Trip Distribution

transportation planning. Murchland [2] seems to have been the first to show explicitly that the gravity model can be formulated as the solution of an equivalent maximization problem and subsequently many authors have formulated various entropy maximizing models [3]. It is convenient to introduce the concept of network entropy in terms of the dimensionless quantities p i j obtained by dividingXj by v as in (13), Sect. 20. The quantity pij is interpreted as the joint probability of a trip being produced at zone i and attracted to zone j, implying the constraints

0 < pij

< 1,

c p i j = 1. iJ

The network entropy is then defined as

H = -xpijInpij.

(3)

i,i

This definition is the same as that used in thermodynamics and information theory and the familiar conventions will be adopted here, e.g., pijlnpij is taken as zero when pij = 0. A single term in the summation in (3) has the following properties: (4)

0, I ,

- p i j Inpij = 0,

for p i j

-pijInpij > 0,

for 0 < p i j < I ,

-pijIn pij has a maximum of e-'

-pijInpij

is concave,

=

when p i j

=

(5)

e-',

for 0 < p i j < 1.

(6) (7)

Properties (6) and (7) follow from d

-( -pij In p i j ) =

dPij

-In p i j - 1 ,

and

The network entropy is a measure of uncertainty, and entropy distribution models are based on the principle that an equilibrium distribution maximizes the entropy. The significance of this principle will be illustrated by a simple example which hardly merits description as a distribution model.

22.

123

Entropy Models

Suppose there is no knowledge about the distribution of trips for a network with m origin and n destination nodes. Then the entropy model is obtained by maximizing the network entropy (3) subject to (2). It is instructive to do this by using a Lagrange multiplier v and defining a Lagrangian function Y(Pij, V ) = -C PijInpij -

~ (Pij2

1).

(10)

Extreme values of this function correspond to

and

a

=o,

-9=---pjj+1 av

which, when solved for the equilibrium values p z , give the unique solutions p; = I/mn

H* = H,,,

=

lnmn.

(1 3) (14)

It is essential to check that the equilibrium values p; given by (13) satisfy the nonnegativity restrictions (1). This example illustrates the technique used in maximizing network entropy; the result is obvious. In the absence of any knowledge, maximizing network entropy distributes the trips evenly through the network, the expected number of interzonal and intrazonal trips being all equal and independent of the zones. ( 6 ) Proportional Model

One of the simplest distribution models is the proportional model which is obtained by maximizing the network entropy H = - x p i j l n pij subject to the restrictions

in addition to the constraints

124

IV. Trip Distribution

The quantities ui, u j are the nonnegative normalized trip productions and attractions defined by (14) and (15), Sect. 20. The model ignores any interzonal trip costs. (i) LAGRANGE MULTIPLIER DERIVATION To obtain the equations defining the model, m Lagrange multipliers, and n multipliers, p j , are introduced, as well as the multiplier v, to define the Lagrangian function

,Ii,

Extreme values correspond to

a

-9 alz,

=

-cpij j

+ ui = 0,

U a~ = - c p i j + uj = 0, a -9 = -c p i j + 1 = 0.

-9 j

i

av

i.i

The solution to (18) is p $ = exp(- 1 - v - A i - p j ) ,

(22)

with the remaining equations forcing ui = exp(- 1- v-di)

Ci exp(-pi),

uj = exp(-l-v-pj)Cexp(-Ai), i

1

=

exp(- 1 - v)

(23) (24)

C exp(-li-pj). i.i

The multipliers are eliminated by taking the product of (23) and (24) and dividing by (25) to give uiuj = exp(- 1 - v-di-pj)

= p*ij .

(26)

125

22. Entropy Models

In terms of the unnormalized quantities (and dropping the asterisk) we have

fij

=

aibj/u.

(27)

The proportional model is characterized as an even distribution of trips subject to given productions and attractions. Jn the absence of any knowledge of the geography of the network--e.g., of interzonal distances-the proportional model maximizes the entropy giving an equilibrium distribution in which the trips between zones are distributed strictly in accord with the proportion of the trips produced at and attracted to the zones. The joint probability of a trip being produced at zone i and attracted to zone j is expressed as the product of u,, the probability of a trip being produced at i, and u j , the probability of a trip being attracted to j . (ii) A PROBABILISTIC INTERPRETATION We now rederive (26) using a purely probabilistic argument which gives further insight into the entropy concept. Let xi denote the probability that origin i is chosen by a single traveler and y j the probability that destination j is chosen. Assume that the choice of origin and destination are made independently of one another. The joint probability of choosing origin i and destinationj and hence the trip ( i , j ) is therefore xiy j . When ai, bj, and u are all integers, the joint probability that al travelers independently pick origin I , ...,a, travelers independently pick origin i, ...,bj travelers independently pick destination j, ..., and so on, is 9; ... $1 ... -frnmy;t ...y y ...Y>

- [x;l X?

...x;my;l ...yv,"]",

(28)

since a, = uui and bj = uuj. In other words, this joint probability is the uth power of a probability P ( x , y ) = x " l x x " z ' ~ ~ ~ x* ~ - y~: ,y ; ' y ~

(29)

which is another way of saying that the joint probability in (28) is the product of u terms, i.e., the probability of the simultaneous realization of u independent events whose probability of occurrence is given by (29). Notice that the probability of indistinguishable arrangements which are obtained by interchanging travelers between different nodes and

126

IV. Trip Distribution

adding terms of the form of (28) as many times as there are interchanges which lead to the same sequences (u1,a2,..., a,,,)(b,,b,, ...,b,) is not the probability that we discuss. Of course, some trip from i or t o j must be chosen by each traveler so that x1 = 1, c j y i = 1. This implies that the sum of joint probabilities of an ( i , j ) trip must equal one, or

xi

c x i y j = 1. i, i

We will only be interested in strictly positive values of xi and yi that satisfy (30); otherwise, (28) and (29) are zero. We shall now show that if there exist xi and y j that satisfy the probability constraint of (30) and, furthermore, if there exists pij that satisfy the conservation laws, (15), then we shall always have -H(p) =

c piilnpij 2 c u i l n x i + i.i

I

i

vjlnyi

=

lnP(x,y),

(31)

or, P ( X A d expC-H(p)l,

for feasible xi, yi, and pii. Furthermore, the equality is attainable in the sense that for feasible, xi, y j , and pij, we have

P

where it is understood that x, y, and p are constrained as before. In order to obtain this result, we make use of a well-known inequality relating arithmetic and geometric means: for arbitrary positive numbers uk which satisfy the normality condition a,

+ a2 + ... + u p = 1,

(33)

and for arbitrary nonnegative numbers zl, z2, ..., zp, we have

a,z1 + a2z2

+ ... + upzp 2 z"1zz"z'

.**

z?,

(34)

with equality holding if and only if z1 = z2 = = zp. This geometric inequality can be written in a slightly more useful form for our purposes, by taking the logarithm of both sides of (34). This yields the inequality 1..

127

22. Entropy Models

since we know that h a > In b, if a 2 b > 0. By writing this expression in terms of new variables tk = Inakzk = Inak + Inzk,

(36)

we now obtain

In [;exp(zk)]

2

(37)

k ak(tk-lnffk)?

or, alternatively,

for arbitrary real f k or arbitrary nonnegative zk. We are now in a position to show that the inequality (31) holds for feasible x,y,p. The right-hand side of (31) is InP(x,y)

=

x

l i

1

uiInxi+C u j l n y j

(39)

Since we assume that the pij on the left-hand side of (31) are feasible, we can substitute the left-hand sides of (15) for ui and uj in (39). This yields the expression

Since each pij premultiplies both a Inxi and a lnyj term, we can collect xiyj products to obtain InP(x,y) where

=

1PijIn(xiyj) = 1pijzij, i j

(41)

i.i

zij = ln(xiyj)

<0

defines zij for all nonnegative xi,yj. We now make use of (38) by substituting zij for the typical tk term, substituting pij for the typical ak term, and using a double rather than a single summation. Since the logarithm of (30) is

'1

1

In C x . y . = In xexp(zij) Fi,j

I

[ij

=

In(1) = 0,

we can write the right-hand side of (41) in the form -InP(x,y) = -C pijzij iJ

I

+ In C exp(zij) , [i,j

(43)

128

IV. Trip Distribution

and make use of the inequality of (38),

to obtain the desired inequality (45)

-lnP(x,y) 2 H(p).

It is now trivial to show that the maximum and minimum values in (32) are actually attained when yi* = v-' bj,

xi* = v-' a,,

p c = uiuj.

(46)

Notice, first of all, that these values are feasible for the probability constraint and the flow conservation equations. Hence, we know that the inequality of (31) must hold. But the entropy is ~ ( p *= ) -

whereas

1uiv j In(uivj),

(47)

i,i

-1uiInui - C vjInvj i = -1uivj(lnui+lnvj)

-inP(u,v) =

i

i.i

=

-Cu,vjln(uivj)

=

H(p*),

LJ

(48)

and the equality has been demonstrated.

PROPERTIES (iii) MODEL It is evident that the proportional model satisfies the conservation requirements, and that the trip numbers are nonnegative, though not necessarily integral. In a practical application, round-off to integers would be satisfactory. The proportional model also has the compressibility property as is easily verified by comparing (4)-(7), Sect. 20, with the following:

ai'bi/v' = aibjlv, a:- bj'/vr = an- bjlv ai' bA- Jut = a, b,,-

Jv

with v' = v,

+ a,,bi/u, + ai b,,/v,

aA-lbA-l/vt = a , - , b , - , / v + a,-,b,,/v+ a,b,,-,/v+a,b,/v, for i , j = 1,2,..., n-2.

(49) (50)

(51) (52)

22.

129

Entropy Models

The model is also separable. Again this can be checked by comparing (12), Sect. 20, with the following:

-

(ai-aibn/u)(bj-an bj/u) u- a,,- bn a,,b,/u

+

- a,(v-

-

6,) bj(v-a,,)

~ ( 0 -an)

(u- 6,)

= aibj/u, (53) for i , j = 1,2,...,n-I. Consider as a numerical example a four-centroid traffic desire network with the following data: Centroid

i

1

2

3

Production Attraction

a,

25

25 20

15 30

bt

40

4

-

__

35 10.

The total number of trips is a, = C bi = 100, so that p i j will be equal to percentages. The trip distribution matrix obtained from the proportional model is, by (27), 1 2 3 4 a,

-

1

10 5

7.5

2.5

2

10 5

7.5

2.5

25

6 3

4.5

1.5

15

Chjl = 3 4

6,

. 14

7

40 20

10.5 3.5

30

10

-

25

(54)

35 100 = U.

To verify compressibility, suppose that zones 3 and 4 are combined. The third and fourth rows and columns of this matrix are added together to give the new distribution matrix I 2 3 ai' (55)

b,'

40 20 40

100 = u'.

130

IV. Trip Distribution

It can be readily verified that for this compressed matrix,

f;i

(56)

= ai' bj'/v',

as given by the proportional model. To verify separability, suppose that zone 4 is removed giving the distribution matrix 1 2 3 a:

(57)

26 13 19.5

bj'

58.5 = v'.

Again it can be checked that f! I J. = ai' bj'lv',

(58)

as given by the proportional model. The proportional model is a simple distribution model which possesses all the desired properties (except integrality of trip numbers). It is of little practical importance for distributing trips, because it takes no account of the geographical disposition of the zones of the study area. (c)

Mean Trip Length Model

Travel patterns indicate a general preference for shorter trips than longer trips and the trip length frequency distributions obtained from travel inventories have proved a reliable characterization of travel in a particular study area. The frequency distributions differ for differing purposes, longer trips being more common, for example, for recreation trips than for home-based work trips. To add more realism to the proportional model, we include just one parameter of the trip length frequency distribution, namely the mean trip length, and we fix this as an additional constraint on our distribution. As has been noticed previously, trip ''length'' is sometimes measured in terms of distance, sometimes as travel time, and we continue our practice of using the more general term cost. If cij is the given fixed cost of a trip interchange produced at node i and attracted to nodej, the total network cost is

c = cJ;jcij i,i

(59)

22. Entropy Models

131

and the mean cost of a trip can be defined as c =

c pii cii . U

(60)

We now impose this relation, with c regarded as fixed, as a constraint on our distribution model in addition to the given productions and attractions. In detail, this entropy model can be formulated as : maximize H subject to

-xpijInpii,

=

i,i

(61)

Ci Pij = ui, C Pij =

(63)

CPij

= 1,

(64)

Ci , i p i j c i j = c.

(65)

uj,

i

t,i

(62)

Such a model has been considered by Sasaki [4] and Tomlin and Tomlin c51. In the usual way, a Lagrangian is first defined as 2(Pij,

pj, V , Y)

which is the same as (17) except for the additional term with multiplier y. The optimal values p z are given by p$ = exp(- 1 - v - l i - p j - y c i j ) ,

(67)

ui = exp(-l-v-li)Cexp(-pj-yycij),

(68)

where i

uj =

1exp(-li-ycij),

(69)

exp( - li-p j - ycij),

(70)

1cijexp(-li-pj-ycij).

(71)

exp(- I - v - p j )

I = exp( - 1 - v ) c = exp(- 1 - v )

iJ

i, i

i

132

IV.

Trip Distribution

It is evident from (67) and (68) that 0 < p c < 1, provided the multipliers are finite. It is not possible to eliminate the multipliers and obtain an explicit expression for p ; ; instead an iterative or model calibration procedure is used. Equation (67) has the form p IJ. . = x .I w11. . yJ.'

(72)

where

wii = exp(-ycii), (73) and the asterisk has been dropped. The x i ,yj, y have to be determined from the constraints wijyi = ui,

(74)

c x i w i j y i = uJ.'

(75)

i

xi

i

cij xi wiiyj

= c.

i,i

The iterative procedure begins with a trial value of 7 from which wii is determined from (73). A convenient starting point is to take the initial value of y as c-'. Then successive values of yi and xi are chosen in order to satisfy (74) and (75). Starting values y j l ) are first chosen (say y$" = ui) and x ! ' ) determined from

in order to satisfy (74). Next y?) is determined from

and, in general,

giving the kth iterate p!k' = X !1k ) w. .yw. 1J 1J J

22.

133

Entropy Models

Termination of this procedure can.be programmed in several ways. One can require, for example, that for some small number 6, the iteration ends when (82) max[lp{t’-p!t-l)l] < 6. (iJ)

When this iteration has been completed, (76) has to be checked and a new value of y chosen in order to reduce any discrepancy. With this new value, the iteration process is carried through again. The calibration of the model to fit the given mean trip length is a rather trial-and-error process and rapid conversion to a satisfactory model requires considerable judgement in gauging the effect of the given distribution of interzonal costs. The justification for the iterative procedure in determining the xi and y j is found in a theorem by Sinkhorn [6] which states that the procedure converges to a unique solution. An important requirement is that the values of wi,imust be positive, which is certainly satisfied in this model. The theorem statement also allows for the obvious fact that the solution for xi and y j is unique up to an arbitrary factor; multiplying xi by a constant and dividing y j by this constant obviously leaves the result unchanged. As a distribution model, the mean trip length model satisfies the conservation equations and its elements are positive (although not necessarily integral). References [4] and [ S ] contain interesting applications of the model to actual data.

(d) Gravity ModeC The gravity model has been the most widely used distribution model in transportation planning and has formed the basis of traffic predictions for many cities. It is not the purpose here to describe the model in great detail but rather to emphasize its relation to the network entropy concept. The gravity model can be regarded as a logical extension of the mean trip length model. Instead of fixing just one parameter-the mean-of the trip length frequency distribution, the complete distribution is fitted. For simplicity we shall let p ( t ) be the percentage of trips with trip lengths cij in the range t < cij < t + At, where At is a suitably chosen increment. For example, cij may be measured in minutes and the trip length frequency distribution represented by a histogram with At = 2 minutes. We write ~ ( t =) pij, (83)

C’

134

IV. Trip Distribution

where the summation C’ is over all pij for which t < cij < t + A t . For each value o f f a Lagrange multiplier y ( t ) is defined so that, in generalization of (66), the Lagrangian function becomes g ( p i j , Ai, p j , V,y ( t ) )

As in (67)-(76), the maximum entropy principle leads to equilibrium values of p 1J. . = x 1. w13. . yJ .’ (85) where wij = exp(-y(t)),

(86)

and xi, y j , and y ( t ) are determined from

c x i w i j y j= uj, i

C’ x i w i j y j = p ( t ) .

(89)

The function exp( - y (t)) is sometimes called the deterrence function and may be chosen as a simple form with parameters determined from (89) or given by a table of values. By analogy with Newton’s law, the deterrence function was originally chosen as 1/ t 2 , corresponding to y ( t ) = 2 In t. However, improved fits were obtained by taking y ( t ) = a t + h I n t with a and h as parameters to be determined by calibration. With this sort of generalization, the model no longer resembles Newton’s law of gravity, and the distribution model is often called an interactance model. In Table 22.1 we illustrate possible forms of the deterrence function. The data in the second column are measured values for home-based work trips for Washington, D.C. [ 123, with t the travel time in minutes. The third and fourth columns of the table illustrate the fit one can get with y ( t ) = 2 In t and y ( t ) = 0.14t, respectively. Suitable multiplicative factors have been used for comparison with the measured values, which are only determined to within an arbitrary factor.

22.

135

Entropy Models

TABLE 22.1 DETERRENCE FUNCTION FOR GRAVITY MODEL (min)

Deterrence function

5 10 15 20 25 30 35

1000 500 205 120 82 53 34

45 50 55

3

I

40

19 9 1

2000 x 5000r ~ 2 exp(-0.14r) 2000 500 220 120 80 55 41 31 25 20 16

1000 500 240 1 20 60 30 15 7

4 2 1

In practice, the calibration of the gravity model is an intricate procedure which, especially for large networks, becomes more of an art than a science. Balancing the row and column sums to satisfy (87) and (88) is achieved by a matrix scaling method similar to that already described. The deterrence function is represented by travel-time or friction factors, called F-factors, and these are modified by special zone-to-zone adjustment factors, called K-factors, until a satisfactory fit with the trip length frequency distribution is obtained. The procedure is rather heuristic, and the model structure tends to become obscured. A difficulty which may arise from zero or small elements in the trip table is the oscillation and nonconvergence of the scaling procedure. This phenomenon is now understood as pointed out in the discussion to reference [ 6 ] . It is interesting to transform the notation used here to the familiar formulation of the gravity model. For a particular trip purpose, the gravity model is usually defined by

136

IV. Trip Distribution

where

A(/')= elements of the trip table at the kth iteration, Fij

=

travel-time factors,

K i j = adjustment factors. If we now define

fg) = up$$', FI.1. KIJ. . = u w i j , by' = y w

1'

then (97) with

and from (94)

I-'

= u j [ I~ X ! k - % v i j

.

(99)

These are precisely (79)-(81). Like the mean trip length model, the gravity model satisfies the conservation equations, has nonnegative (but not necessarily integral) elements, but does not possess the compressibility or separability properties. Despite its almost universal use, the model is now regarded by some with suspicion [S]. 23.

Opportunity Models

Opportunity trip distribution models come second only to the gravity model in popularity among transportation planners. Stouffer [9] first introduced the concept of opportunities as a useful variable in describing travel behavior and applied the concept to the residential mobility of people.

137

23. Opportunity Models

For a traffic desire network, the opportunities are determined as possible destinations. For a particular origin zone, the intervening opportunities are calculated from an ordering of all destination centroids according to their costs (distances, travel times etc.) from the origin. If, as in our notation, we define bj as the number of trip destinations at centroid j , then the number of intervening opportunities between (and including) origin i and destination j is given by

B.. IJ = C ' b , ,

(1)

where C'is the sum over all centroids I for which cil < cij. We shall describe two trip distribution models using the concept of opportunities. (a) Intervening Opportunities Model

The intervening opportunities trip distribution model was developed for the Chicago Area Transportation Study (CATS). The model is based on the following assumption : the conditional probability that a trip having originated at zone i has a destination beyond j is given by exp( - Li Bij), where Bij is the number of intervening opportunities between i and j, and Li is a constant. This exponential distribution is familiar in the kinetic theory of gases (when the probability refers to a molecule traveling a distance without colliding), and the theory of radioactive decay (when the probability refers to an atom not decaying in a certain time). The quantity l/Li,analogous to the mean free path between molecular collisions or mean life of a radioactive atom, is interpreted as the mean number of intervening opportunities not accepted. The estimated trip numbers from i beyond j are given by the formula

C A, = aiexp(-LiBij).

(2)

czl> CiJ

The actual trip numbers between i a n d j are obtained by differencing, or more explicitly, as

fij

=

2' b,)] ,

ai [exp( -L~C" b,) - exp( -L~

(3)

where C"is the sum over all centroids 1 for which cil < cij, i.e., without the equality sign used in determining This is the mathematical statement of the intervening opportunities trip distribution model. It is evident that thehi are positive (although

x'.

138

IV. Trip Distribution

not necessarily integral), but that the model does not satisfy the conservation equations Cfij

i

=

ai,

Cfij = b j . i

(4)

(5)

In practice, (4) will be approximately true, and the row balances implied by (5) are achieved by an iterative scheme

similar to that used for the gravity model (see (90)-(92), Sect. 22, and also Problem 7, Sect. 27). The final calibration of the model requires the choice of appropriate L-factors Li which may differ from zone to zone and vary for different purposes. In fact, there is evidence from opportunity curves drawn from data for various cities that the linear term in Bij appearing in the exponent in (2) is inadequate, and the curves are better represented by a cubic. From the complexity of the intervening opportunities model, it is evident that as a trip distribution model it does not possess the compressibility or separability properties.

(b) Preferencing Model An opportunity trip distribution model which satisfiesthe conservation equations without any iterative balancing has been developed by Kirby [lo] using the preference concept familiar in utility theory. Each origin zone is considered to rate the available trip destinations in order of preference, according to interzonal costs. The problem of determining an optimal trip distribution for given preferences is then analogous to the marriage problem [l 11 in which each man of a group rates a group of women in order of preference, and an optimal pairing of couples is sought. To describe the preferencing model, we first consider our network as a “blown up” bipartite network with u origin nodes and u destination nodes

23.

139

OpportunityModele

with one trip originating from each origin zone. The distribution of trips then corresponds to a pairing (or marrying off!) of the origin and destination zones. Each origin i rates the u possible destinationsj in order of preference, and likewise each destinationj orders the possible origins. A trip distribution is a 1-1 mapping s of the origins onto the destinations so that s ( i ) is the destination of the trip originating from i. A stable trip distribution has the following property: if origin i prefers destination s(h) to s ( i ) , then destination s(h) prefers origin h to origin i. The distribution would be unstable if i prefers s(h) to s(i), and s(h) prefers i to h, for then a swap of destinations would be mutually preferable. In general, there are many stable distributions, and among these we seek distributions which are optimal. It is necessary to distinguish between origin optimal and destination optimal. An origin-optimal distribution is a stable distribution for which each origin attains its highest preference destination consistent with stability. More precisely, s is said to be origin optimal if, for any stable distribution r, each origin i prefers s ( i ) (or is possibly indifferent) to r ( i ) . Likewise for a destination-optimal distribution, each destination attains its highest preference origin consistent with stability. Interesting results in preference theory imply that there always exists just one origin-optimal distribution and just one destination-optimal distribution, and if a distribution is both origin optimal and destination optimal, it is the only stable distribution. The following example will illustrate these ideas. Suppose that there are u = 7 trips to be distributed between 7 origins and 7 destinations, and suppose that the origins rate the destinations in the orders of preference in Table 23.1. The elements of this table are interpreted as follows: origin 5 rates destination 7 as first preference, destination 6 as TABLE 23.1 ORIGINPREFERENCE Destinations .

.

1 2 3 4 5 6 1

_

5 1 5 1 5 2 6

_ 3 3 6 5 3 3 5

3

5

4

4 ~ 2

2

6

1

7 6 4 1 1

3 1 6 6 4

4 2 1 7 3

1

~

6

1

7 4 2 4 2 5 1

6 5 1 3 1 4 2

140

IV. Trip Distribution

second preference, destination 2 as third preference, and so on. Likewise, suppose that the destinations rate the origins in the orders of preference given in Table 23.2. The algorithm for determining the origin-optimal TABLE 23.2 DESTINATION PREFERENCE

.z8 .o_2

Origins

Q s l

2

~~

1 2 3 4 5 6 7

7 2 1 6 4 2 6

5 1 2 5 5 1 7

3

-~

6 3 3 7 3 3 5

4

5

6 ~

3 7 6 3 2 4 1

4 6 7 4 1 5 2

~

2 5 4 2 7 7 4

7 L

._ _

_

1 4 5 1 6 6 3

distribution, which incidentally also provides a constructive proof of the existence and uniqueness of this distribution, proceeds as follows: (i) the origins are first assigned their top rated destinations; (ii) the line of destination numbers is checked to see whether any number is repeated-if not, the origin-optimal distribution has been reached-if there are repeats, proceed to (iii); (iii) take any pair of repeated numbers and from the destination preferences retain the number which has the higher preference and change the other number to the number corresponding to the next lowest destination preference; (iv) return to Step (ii).

Thus, (i) gives the first line in Table 23.3. Since 7 is repeated, the preference ordering for destination 7 is checked from Table 23.2, and origin 5 has a higher rating than origin 3, so the trip from origin 5 to destination 7 is retained. The destination preferences for origin 3 are checked and instead of destination 7, destination 6, which is the next lowest, is assigned. This gives the second line of Table 23.3. The procedure is repeated until the final line in the table is reached, signifying

141

24. Combined Distribution and Assignment

TABLE 23.3 ORIGIN-OPTIMAL DISTRIBUTION Origins 1 __ 5 5 5 5 5 5 4 2

2

3

4

5

6

7

7 6 6 6 6 6 6 6

4 4 4 4 4 5 5 5

7 1 7 7 7 1 7 7

3 3 3 3 3 3 3 3

6 6 7 5 4 4 4 4

.~

1 1 1 1 1 1 1 1

-

-~

-~

all destinations assigned and the origin-optimal destination. It is instructive to check that this distribution is stable. For example, using the notation that origin iis paired with destination s ( i ) , s(l) = 2 and 4 7 ) = 4. Origin i = 1 prefers destination 4 to destination 2, but destination 4 prefers origin 7 to origin 1. To apply this procedure to the practical problem of trip distribution, it is necessary to group the origins and destinations into zones and replace individual trip preferences by group preferences. For calibration purposes, the group preferences are determined from opportunity curves. By an extension of the algorithm described above, an originoptimal or a destination-optimal zone-to-zone trip distribution is obtained. The number of tripsAj from origin zone i to destination zone j is the total number of individual trips originating from the group ai origins in zone i which have destinations within the group of bj destinations in zone j. The trip preferencing model automatically yields nonnegative integral trip numbers which are correctly balanced, satisfying the conservation equations. The model is neither compressible nor separable. The reader is referred to [lo] for an account of the application of the model to actual data. 24. Combined Distribution and Assignment

It is customary in the transportation planning process to consider traffic assignment separately from trip distribution, but it is important to realize that the two are dependent on each other. The deterrence

142

IV. Trip Distribution

function used in trip distribution models depends on the congestion and traffic flow on the networks, and the output of the assignment model should serve as a n input to the distribution model. This feedback is difficult to incorporate into the planning process, but its neglect is likely to be highly significant. Several attempts have been made to combine distribution with assignment, and some of the methods which have been developed will be discussed below. The simplest resort to repeated use of distribution and assignment programs. ( a ) TRC Program

For a transportation study of Toronto, Canada, Traffic Research Corporation [ I61 developed an iterative procedure which specifically allowed for the feedback between distribution and assignment on the basis of a n assumed cost-flow relation. At the end of the first pass of the process, the link flows were obtained by all-or-nothing assignment and the link costs updated. For the second iteration, new cheapest route trees were built, with new interzonal costs determined for input to the distribution model. The O-D traffic was then assigned in some proportion to the two cheapest routes already determined. The whole iteration was repeated until convergence was obtained. The program allowed up to 9 routes between each O-D pair.

(6) LTS Program The analytical methods developed for the London Transportation Study [ 171 allowed for feedback between congested assignment and trip distribution as follows. All-or-nothing assignment was first used to indicate where the traffic demand exceeded the network capacity. The speeds which had been assumed for the overloaded links were then reduced until all roads were equally overloaded. Two successive speed changes were needed to achieve this balance, and each time the speeds were altered, the trips were redistributed. A more drastic technique achieved a reduction in trips in order to accommodate the traffic. This was assumed to correspond to some form of restraint on trips by regulation or pricing. The multiple route assignment described in Sect. 14(d), Chap. 111, has also been used in an iterative distribution-assignment scheme. Its

25.

143

Conclusion

application in the LTS has indicated that it forces a fairly rapid convergence and that, except when the initial overloading of the network is severe, redistribution of the trips is only necessary on alternate passes. ( c ) Multicommodity Distribution-Assignment From the theoretical point of view, the most satisfactory method of combining distribution with assignment is to incorporate them in the one minimum network cost multicommodity flow model. The combined model requires some extension of notation. We denote centroids by k = 1,2, ...,n, and an origin-destination pair by k,k‘. The trip productions and attractions, which usually form the input to the distribution models, are denoted by ak and bk’.The flow of commodity kk‘ on thejth chain from k to k‘ is denoted by hgk’ with corresponding route cost C y ’ . The minimum network cost problem is then represented by the program h y ‘ 2 0, k‘j

~~~

k k ‘ j

hgk’

=

d,

Cy’hgk‘ = C(min).

(4)

Link capacities can be introduced if required. Although this model is attractive from the theoretical point of view, its practical use is limited because of the difficulties in solving the mathematical program for large networks with flow dependent link costs. Nevertheless, some interesting calculations have been carried out and reported [ 181. 25.

Conclusion

In describing various trip distributions, the emphasis has been placed on their basic structures and properties. It has been shown where the extremal principles analyzed in the previous chapter are inherent in the models. Except for the models which are widely used and are well

144

IV. Trip Distribution

documented with users’ manuals, no attempt has been made to give practical details of how the models should be programmed for computer analysis. Transportation planners are concerned with the choice of the distribution model best fitted to their purposes. Comparative evaluations of the different models have been made by various authors [12], [13] and these together with other critical reviews [S], [14] tend to be rather inconclusive and underline the necessity for a better understanding of the model structures. An interesting comparison of the Hitchcock and gravity models has been made for a theoretical town [ 151. The preferencing model has been described in some detail because of its novelty. The present trend in research tends to favor the further analysis of opportunity curves because of their usefulness in calibration and prediction. 26. Notes and References

Hitchcock, F. L., The Distribution of a Product from Several Sources to Numerous Localities, J . Math. andPhys., 20,224-280 (1941). Because of the rather confused history of the development of linear programming and its applications, it is probably wisest to avoid attaching specific authors’ names to models and problems, but the transportation problem is often called the Hitchcock problem although the double barrelled Hitchcock-Koopmans epithet is preferred by some. This early paper clearly formulates the model. Although the given solution is somewhat sketchy, its simplicity is still attractive and provides an excellent introduction to the classical transportation problem. [l]

Murchland, J. D., Some Remarks on the Gravity Model of Traffic Distribution and an Equivalent Maximizing Formulation, Report LSE-TNT-38 (1966). This short excellent report contains some interesting comments on the gravity model and its formulation as a maximization problem. [2]

Wilson, A. G., The Use of the Concept of Entropy in System Modelling, Operations Res. Quart., 21, 247-265 (1970). Some controversy has arisen over the use of the concept of entropy in the social sciences. The point of view expounded in this paper is

[3]

26. NotessndReferences

145

challenged by D. J. White in a short note on pp. 279-281 of the same journal. The point at issue is whether entropy can be used to predict equilibrium distribution patterns in the context of travellers who know where they want to go and select rational alternatives for getting there. It is agreed by various authors that the entropy concept is helpful in formulating trip distribution models-but why it works is the question. M. J. Beckman and T. F. Golob, argue in a paper (On the Metaphysical Foundations of Traffic Theory : Entropy Revisited, presented at the Fifth International Symposium on the Theory of Traffic Flow and Transportation, held at the University of California, Berkeley, in June, 1971) that the trip distribution formulae are better derived from a utility maximization model for which the basic assumption is that “households are assumed to behave rationally (i.e., to maximize net utility).” The paper by Wilson includes an extensive list of references, among them Vol. 14, No. 1, (1970) of Transportation Res., an issue devoted solely to trip distribution models. Sasaki, T., Probabilistic Models for Trip Distribution, Proceedings of the Fourth International Symposium on the Theory of Traffic Flow, Karlsruhe (1968). Professor Sasaki has written a series of papers on the application of Markov chain theory to trip distribution, and the present paper exploits the entropy maximization method. The distribution model is applied, with numerical results, to shopping trips in Kyoto City and to the trip distribution between ramps on the Hanshin Expressway. [5] Tomlin, J. A. and Tomlin, S. G., Traffic Distribution and Entropy, Nature, 220, 974-976 (1968). This son-father paper uses the theory of statistical mechanics to formulate distribution models in terms of the entropy concept. A model analogous to the fixed mean trip length model is described and checked with real data from the Metropolitan Adelaide Transportation Study. A good fit to the data is obtained with y = l/c, in the notation of (66). Sect. 22. [6] Sinkhorn, R., Diagonal Equivalence to Matrices with Prescribed Row and Column Sums, Amer. Math. Monthly, 14, 402-405 (1967). The author proves the following: [4]

“THEOREM : Let rl ,...,r,, c, ,...,c, be fixed positive numbers. Then, corresponding to each positive m x n matrix A, there is a unique matrix

146

IV. Trip Distribution

of the form D, AD, with row sums pr,, ...,pr, and column sums c,, ...,c,, where p = cj/Ciri. D, and D, are respectively m x m and n x n diagonal matrices with positive diagonals and are themselves unique up to a scalar multiple. The iterative process of alternately scaling the rows and columns of A to have row and column sums respectively ri and cj can be used t o j n d D, AD,. The subsequence from the iteration, in which column sums are scaled, converges to D, AD, while the subsequence in which the row sums are scaled converges to (l/p)Dl AD,. In particular i f x i r i = C j c j then the entire iteration converges to D, AD,.

xi

In an earlier paper to which the author refers, examples are given to illustrate the possible breakdown of the iterative process when the matrix A has zero elements. It is also shown that replacing the zero elements by small positive quantities, iterating, and then taking the limit does not lead to unique results. Tanner, J. C., Factors Affecting the Amount of Travel, Road Res. Lab. Tech. Paper, 51, Her Majesty’s Stationery Office, London (1961). In this important paper, Tanner discusses possible deterrence functions in considerable detail. He points out that when using t (time or distance) as a continuous variable, so that sums become integrals, a deterrence function of the form t - n theoretically implies infinite total travel time on long trips if n < 3, and infinite total travel time on short trips if n 2 3. The singular behavior of the integrals can be eliminated by choice of a function of the form t -be-afcorresponding to y ( t ) = at b In t in the notation of (86), Sect. 22. [7]

+

Heggie, I. G., Are Gravity and Interactance Models a Valid Technique for Planning Regional Transport Facilities?, Operations Res. Quart., 20, 93-1 10 (1969). This paper concisely defines the gravity and interactance distribution models and then asks the penetrating questions: are the hypotheses reasonable; are the models logically consistent; and do the models fit the facts? The author’s general conclusion from his analysis is that gravity and interactance models do not provide a valid means of producing traffic forecasts in a regional development. In “A Rejoinder” on pp. 489-492 of the same journal volume, A. G. Wilson criticizes Heggie’s paper as offering “an extremely misleading review of gravity and interactance models.” Wilson’s note is followed by a spirited reply [S]

26. Notes and References

147

from Heggie-only to be followed by “A Further Rejoinder” by Wilson. The lively debate on this controversy gives an interesting insight into the gravity models and their applications. Stouffer, S. A., Intervening Opportunities: A Theory Relating Mobility and Distance, Amer. Sociol. Rev., 5, 845-867 (1940). This interesting and readable paper propounds the theory that there is no necessary relationship between the mobility of migrating people and the distance they migrate, but that “the number of persons going a given distance is directly proportional to the number of opportunities at that distance and inversely proportional to the number of intervening opportunities.” This important concept of opportunities is defined by the author in terms of the particular problem being analyzed. Thus “for a white family leaving a dwelling in rental group K i n tract X , the number of opportunities in tract Y is proportional to the total number of white families, whatever their place of origin, moving to dwellings in rental group K within tract Y.” The paper includes an extensive test of the proposed theory with empirical data.

[9]

Kirby, R. F., A Preferencing Model for Trip Distribution, Transportation Sci., 4, 1-35 (1970). This paper gives a detailed account of the theory of the trip preferencing model and a formulation of the algorithm for its practical application. The model is illustrated with an example using data from a transportation study of the city of Launceston, Australia. The paper is a condensed version of part of the author’s doctoral thesis.

[lo]

[ll]

Gale, D. and Shapley, L. S., College Admissions and the Stability of Marriage, Amer. Marh. Monthly, 69, 9-15 (1962). This entertaining paper describes in a particularly lucid manner the basic ideas of a preference model for the distribution, not of trips, but of students to colleges. The model is also applied to the problem of determining stable and optimal marriages in a community ! Although the structure of the preference model is easy to describe in a loose nonmathematical way, the precise mathematical formulation of an algorithm (given in the previous reference) is quite complicated. Heanue, K . E. and Pyers, C. E., A Comparative Evaluation of Trip Distribution Procedures, Public Roads, 34, 43-51 (1966). This important paper reports on a research project designed to test and evaluate the following trip distribution models: Fratar, gravity, intervening opportunities, and competing opportunities. The validity

[12]

148

IV.

Trip Distribution

of the models for forecasting purposes is analyzed by comparison with a seven-year historical period for Washington, D.C. The project concluded by rating the gravity and intervening opportunities models about equal and somewhat better than the other two.

Lawson, M. C. and Dearinger, J. A., A Comparison of Four Work Trip Distribution Models, J . Highway Div., Proc. Arner. SOC.Civil Eng., 93, 1-25 (1967). This paper reports the results of a comparison of four trip distribution models : electrostatic, gravity, competing opportunities, and multiple regression. The comparison is made for work trips in Lexington, Kentucky. Various suggestions are made for improvements to the models although the gravity model is favored as giving the best results.

[13]

Fairthorne, D. B., Description and Shortcomings of Some Urban Road Traffic Models, Operations Res. Quart., 15, 17-28 (1 964). This paper gives a lucid description of the gravity and opportunity models and is particularly valuable for its analysis of the theoretical bases of these models. The author, while admitting that the ultimate test of a model is the accuracy of its predictive power in practical applications, suggests that a useful practical model is likely also to be one which is theoretically consistent. The inconsistencies and shortcomings of the gravity and opportunity models are revealed in some detai 1.

[14]

[lS]

McDonald, W. R. and Blunden, W. R., The Application of Linear Programming to the Determination of Road Traffic Desire Line Patterns, Proc. Australian Road Res. Board Conf., 4, 153-168 (1968). This paper compares the Hitchcock trip distribution model (referred to simply as the linear programming model) with the gravity model for the hypothetical city Lautsville, developed by the Los Angeles Regional Transportation Study as a test city for illustrating basic principles involved in forecasting travel. The authors give cogent reasons for the usefulness of the Hitchcock model, especially for planning purposes. Irwin, N. A. and Von Cube, A. G., Capacity Restraint in Multi-Travel Mode Assignment Programs, H.R.B. Bull., 347, 258-289 (1 962). The TRC programs have been noteworthy for their consistent and valid approach to transportation planning. This paper describes the

[I61

149

27. Problems

basic program blocks-tree generation, time factor, trip distribution, proportional split, assignment and link updating-and explains the interconnection between these. The incorporation of proportional (modal) split within the iterations is described in detail. [I71 Tresidder, J. O., Meyers, D. A., Burrell, J. E. and Powell, T. J., The London Transportation Study : Methods and Techniques, Proc. Inst. Civil Eng., 39, 433-464 (1968). This paper is an excellent summary of the new improved techniques which were developed for the LTS. As can be expected for a conurbation of the size and complexity of the London Study Area, the techniques are principally designed to allow the estimation of flows on a heavily loaded transportation network, with the emphasis on the economic evaluation of alternative transportation plans.

Tomlin, J. A., A Mathematical Programming Model for the Combined Distribution-assignment of Traffic, Transportation Sci.,5, 122-140 (1971). The author illustrates his theory by applying his method to the network used by Charnes and Cooper in [24], Sect. 17, Chap. 111. [18]

27. Problems 1. A traffic desire network has 3 origins, i = 1,2,3, and 4 destinations, j = 1,2,3,4, and the number of trips ai originating from i, the number of trips bj with destinationj, and the costs cij of a single trip from i t o j are given by the table

6,

1

2 3 4 a i

I

6 2 6 15=v.

The solution of the Hitchcock trip distribution model gives the following optimal trip table with the associated implicit prices

IV. Trip Distribution

150 Ui .

~

11

3

0

115

lo

3

2

019

0

0

0

517

bj 0 - 1 -5

-3.

If destination nodes j = 3,4 are combined, calculate the new costs according to (11)-(13), Sect. 21, and verify that the compressed trip table is optimal.

2. Repeat Problem I but with destinations 1 and 2 combined instead of destinations 3 and 4. 3. A traffic desire network with four centroids has productions and attractions as follows : Centroid Production Attraction

1

2

3

4

40

30 20

20 30

10 40

_-10

Calculate the trip table for the proportional distribution model. Verify the compressibility property by combining centroids 2 and 3 and the separability property by removing centroid 2. 4. For the same data as in Problem 1, use the iterative procedure described in Sect. 22 to determine the trip table giving a mean trip length of 5.50 measured in the same units as cij.

5. For a study area with 3 zones, the trip table for the base year is measured to be a!') 7

by)

11 7 7

25

= U.

151

27. Problems

It is predicted that the growth factors to give the estimated productions and attractions for the design year are Growth factors Zone

Production

1

2 3 4

2 3

Attraction

____

-

3 4 2 .

Use a matrix scaling method to estimate the trip table for the design year.

6. A traffic desire network with 3 centroids has productions and attractions as follows : Centroid Production Attraction

1

2

3

14 33

33 28

28 14

.

The inter- and intrazonal travel times as given by a skim trees program are 1 2 3

and the travel times factors Fij are given by the following table:

7

8

Fij 82 52 50 41 39 26 20

13

C ij

1

2

3

4

5

6

.

Assuming K-factors are all unity, use the scaling technique of the gravity model to obtain three iterations of the trip distribution table.

7. A traffic desire network with 3 centroids has productions and attractions as in Problem 6, but the zone-to-zone costs are

[Cij]

=

1:: :I. 2

I

3

152

IV. Trip Distribution

The L-factors for the 3 zones are L , = L, = 0.04, L, = 0.02. Use the intervening opportunities model to determine the distribution of trips.

8. For the same data as given in Tables 23.1 and 23.2, determine the destination-optimal trip distribution. 9. (a) Find the (normalized) flows pij, i = 1, ...,m,J= 1, ...,n, which maximize the network entropy

H =

-C

iJ

pijhpij,

subject to

(ii) (iii)

i

pij = ui ;

Ci pij = uj ;

(iv) pll

=

k.

where ui, vj and k are given positive quantities. (b) Find a numerical solution for the case where m = n = 3, u1 = u2 = 0.25, u3 = 0.50, u1 = u3 = 0.40, v2 = 0.20 and p1 = 0.01.

APPENDIX

THEOREM FOR CHEAPEST ROUTE ALGORITHMS

In this appendix we prove the following

THEOREM: The costs of the cheapest routes from node n , to nodes n, of a network [ N ; L] with positive link costs c(ni,nj)are the unique solutions of the functional equations

Proof: For simplicity, we suppose that the network is connected and undirected, and we adopt the usual convention that c(ni,nj)= 00 if (ni,nj)6 L. Denote the cost of a path from nl to n, by C(n,,n,), and the cost of a cheapest path from n, to nr by C*(n,,n,). Then we have to prove that C*(n,,n,), for given n,, and for all n, E N , are the unique solutions of (1) and (2). (a) The first step in the proof is to show that the cheapest costs C*(n,,n,) satisfy (1) and (2). The cheapest path from n , to n , is the “empty” path without links for which C * ( n l , n , ) =0, so that (1) is satisfied. Suppose n , # n , and let n , , ...,nj,n, be a cheapest path with cost C*(n,,n,). Then C*(n,,nr) = C(n1,nj) + c(nj,nr), 153

(3)

154

Appendix A

where C(n,,nj) is the cost of the path n,, ...,nj. Hence, C*(n,,nr) 3 C*(n,,nr) + c(nj,nr)

3 min [C*(nl,ni)+c(ni,n,)] ni #n,

= C*(nl,nj')

+ C(njt,nr).

(4) (5)

(6)

Relation (4) follows from (3) because C 3 C*, and in (6) nj' is a node # nr, for which the minimum in ( 5 ) is attained. Attention is now turned to a path n,, ...,nit, ...,nj' which has cost C*(n,,nj').We extend this path to give a route n,, ..., n,', ..., nj', n, from n, to nr, and we prove that this route is a path by showing that it does not visit node n, twice. If the contrary were true, so that n,' = n,, say, then n,,..., n,' = n, is a path from n, to n, with cost

C'(n,,n,') < C * ( n , , n j ' ) + C(nj',nr) G C*(nl,nr),

(7) (8)

by (4)-(6). This implies n,, ...,ni' is a path from n, to n, with cost less than C*(n,,n,), which is impossible. Thus, n,, ..., n,', ...,nil, n, is indeed a path from n, to n,, and its cost is

C'(n,,n,) = C*(n,,ni)

+ c(nj',nr)

G C*(n,,nr),

(9) (10)

by (4)-(6). But since C*(n,,n,) is the cheapest cost, the 2 signs in (4) and (5) are equal signs. Inequality ( 5 ) , as it now becomes, proves that the C* satisfy (2). Note also that (4) with an equal sign now implies that n,, ..., nj is a cheapest path from n, to nj. (b) We next prove that the solutions to (1) and (2) define a path from n , to n,. We first observe that iff'(n,), n, E N , is any solution set of ( I ) and (2), then 0
,

f(nr) =f(nr-I)

We call n,-

+ c(nr-,,nr).

, a predecessor node of n,.

(1 1)

155

Theorem for Cheapest Route Algorithms

We next show that for given n,, the solution set of (1) and (2) defines a path n,,nz, ..., nk- ,,nk, ...,n, from n, to n,, such that

As noted above, the solution of (2) defines a predecessor node nr-l of nr and successively a sequence of nodes n,,n,- ..., nk,nk-1, ... with n k - , the predecessor of nk. From ( l l ) , the relation (12) holds for each pair of successive nodes. It is only necessary to show that the sequence of nodes defines a path from n, to n,. This is a consequence of the fact that f(n,),f(n,- ,), ...,f(nk),f(n,..., is a strictly decreasing sequence, which negates the possibility that a node can appear more than once in the sequence, and forces the sequence to reach n , , the only node for which f ( n i ) = 0.

,,

(c) Finally, we show that the solutions to ( I ) and (2) are unique by supposing, to the contrary, thatf, and fzare two solution sets and that for some n, E L, fl (n,)
,,

fl (nk)

- 1 1 (nk-

1)

=

c(nk-

=

2, 3,

..-?

r.

(13)

In particular, fi(nr)

+ c(nr-1,nr)
(14)

since f z ( n , ) is a solution of (2). From (14) it follows that f , (n,- 1) < f 2 ( n , - ,), and the argument is continued until we deduce that f l ( n , )
This page intentionally left blank

APPENDIX

DUALITY THEORY

With notation which is unrelated to that used in the text, a primal linear program and its dual can be written in general form as Primal P xj

2 0,

xi unrestricted in sign,

Dual D m

C ~ i a iG j I

1

cj,

yiaij = cj,

j

=

I , ...,ni, (1)

j = n,

+ 1, ...,n, (2)

i = I , ...,m,,(3)

qixj

I

=

yi unrestricted in sign,

bi,

i

=

m,

+ I , ...,m, (4)

n

C c j x j = z(min), 1

m

C biyi = u(max). 1

The main results of duality theory are: (a) the dual of D is P ; 157

(5)

158

Appendix B

(b) if xi is a feasible solution of P and yi is a feasible solution of D (so that the constraints (l)-(4) are satisfied), then

(c)

if P and D both have feasible solutions, then they have optimal solutions xi* and y,*, and the optimal values of z and u are equal :

(7)

z* = u*.

For optimal solutions, the following complementary slackness relations are valid: xi* > 0,

(d) if

m

then xyi*ai, = ci, j 1

= 1,

..., n,, (8)

(e) if

m

x y i * a i j < ci, 1

then

xi* = 0,

j = 1 ,.'., n,,

(9) yi* > 0,

(f) if

then

n

1 aijxj* = bi, 1

i = 1, ..., mi,

(10) n

(g) if

I

aiixi* > hi,

then

yi* = 0,

i = 1,

..., m,. (1 1)

APPENDIX

INEQUALITIES FOR MARGINAL AND AVERAGE LINK AND CHAIN COSTS

Throughout this book we have used the idea of the average cost of link flow ci(fi),the total cost of a link flowfici(fi), the marginal cost of link flow di(fi),and corresponding average and marginal costs C y ) = Cy)(h)

=

2i ci(fi)a{:),

j

E M(k)

j = 1,2, ..., m(k),

Dp’ = Dy’(h)

=

(1)

2i di(fi)a!:’,

for chains connecting the O(k)-D(k)pair. The quantity a$) is unity if link i is a member of chain m y ) connecting O(k)-D(k),zero otherwise. The purpose of this appendix is to derive certain inequalities for average and marginal costs of link and chain flows given that for the total cost of link flow: (i) f i c i ( f i ) is strictly convex and increasing in [0, co). (ii)

lim f i c i ( f i ) = 0.

h+O+

It should be pointed out that condition (i) implies that .fi ci(fi) is continuous ( [ 2 3 ] , Sect. 17, Chap. 111). In the remainder of this Appendix, and in Sect. 16, we use the following result. 159

160

Appendix C

If 0 < x , < x, < x 3 are three values of x , and y, = y ( x , ) , y , = y(x,), y 3 = y ( x 3 ) are the three corresponding values for the strictly convex function y(x), then the following inequalities hold: Yz-Y, <

Y3-Y1

x,-x1

x3-x1

< Y3-Y2 x3-x,

(3)

In other words, the slopes of the three chords connecting any pair of the three points (x1,y1)(x2,y2)and ( x 3 , y 3 )satisfy (3). Since y ( x ) is strictly convex, we have from its definition ay(z1)

+ (1 -a)y(z,)

’Y(L=,

+ (1 - a ) z 2 ) ,

(4)

for any z2 # zl and every a such that 0 < ci < 1. Consider first the chord connecting (x,,y,) with ( x 3 , y 3 ) and the chord connecting (xl,y,) with ( x 2 , y 2 ) . Since x1 < x 2 < x 3 , there exists a value of a, say E , such that x2 = Ex,

+ (I -E)x3.

(5)

From the definition of a strictly convex function in (4) we obtain y2 = ~ ( x 2 < ) Cly(x1)

+ (1 - E ) y ( x 3 ) = olyi + (1 - @ - ) . ~ 3 .

(6)

Therefore,

It is also easy to show that the right-hand inequality of (3) holds. We can now show that conditions (i) and (ii) on the total cost of flow imply that the average cost of flow, ci(JJ, is monotone increasing. We use the three chords lemma in (3) with y =f;:ci(J;.).Substituting x1 = 0, x2 =fi, x3 =f;‘into (3), yields the inequality

or ci(fi)

< ci(.fi‘),

for 0 < j ;
(8)

Thus, we have shown that conditions (i) and (ii) in (2) imply that ci(f;:)is monotone increasing. To obtain inequalities for marginal costs of link and chain flows, we make use of the well-known mean-value theorem ([23], Sect. 17, Chap. 111):

161

Marginal and Average Link and Chain Costs

if y ( x ) is a differentiable function of x in the open interval (a,b), then there always exists a particular value of x, say z, such that a < z < 6 , and

Geometrically, the theorem states that one can always find a value of x for which the slope (tangent line) at x times an interval of length b - a is exactly equal to the difference in the values of the function evaluated at a and b, respectively. The only requirement on z is that it lies between a and b. I n conjunction with properties (i) and (ii), (9) can be used to prove that the average cost of link flow ci(A)is strictly less than the marginal cost di(A),and that the following inequalities also hold:

L’ci(h’)-A ci (A) > di (fi) (A’ -Ah A’ci(A7 -Aci(fi)

< dicf,‘)U‘-A)*

A’ # A ,

(10) (1 1)

The difference in the two inequalities is simply that the inequality is reversed when the marginal link cost is evaluated at A‘ rather than A. We know, from (9), that if &
.fitci (.&’I -A ci (A) = di(Ti)(A‘ -h>*

(12)

Making use of the fact that for a strictly convex total cost the marginal costs are strictly increasing with flow, di (A) < di (A) < di (A’)9

(13)

we are led to the inequalities of (10) and (1 1) for A A‘. A simple sketch will illustrate the result. It is a simple matter to show that these inequalities carry over to average and marginal costs of chain flows. By summing (8) over all links in a given chain, we obtain

Cjk’(h’)= C ci(fi’)a!:’ > 1 ci(fi)u$’ i

i

=

C(&’(h) J ,

(14)

for any chain where at least one link flowfi’ and all other link flows in the chzin do not decrease. By h’, we mean the flow pattern corresponding to the new link flow vector f’, i.e.,

fi‘

=

k i

.!$’h;(k’.

(1 5 )

162

Appendix C

As an example, if the flow in one chain changes from hfk) to = h(k) + A 9

A>O,

(16)

then at least one link in that chain has an increase of flow equal to A (recall that by our convention in Sect. 7, Chap. 11, two distinct chains connecting an O(k)-D(k)pair must have at least one distinct link). We can also obtain similar inequalities for marginal chain costs. Substituting (10) and (1 1) into (1) gives Dy)(h)A = C di(fi)u::) A i

< C Cfi’ i

ci

(.A’) -.A ci(fi)Ialjk’,

A > 0,

(17)

where hi(k)= hf)+A, and in at least one link&’ =f,+A. Relation (17) simply says that the difference in total costs of all links in thejth chain (due to the new flows) is greater than the marginal chain cost times the increase in flow. The inequality is reversed if either A is negative or the marginal link costs are evaluated at & + A ; the inequality holds if the marginal costs are evaluated at J;: -A.

APPENDIX

D ANSWERS TO PROBLEMS

Chapter II 12.1 No. L does not contain (1,4) or (4, I), for example. 12.2 Yes. 12.3 There are nine chains, listed below as sequences of links: m, = (1,2),(2,3),(3,4),(4,5),(5,6), m2

= (1,2), (2,3), (3,4), (4,6),

m 3 = (1,2), (2,3), (3,5), (5,6), m 4 = (1,2), (2,4), (4,5), (5,6), m s = (1,2), (2,4), (4,6),

m 6 = (1,2), (2,5), (5,6), m- = (1,3), (3,4), (4,5), (5,6), ms = (1,3), (3,4), (4,6),

m g = (1, 3), (3,5), (5,6). 163

164

Appendix D

12.4 There are eleven paths, which are not chains. They are listed below as sequences of links: (i) (1,2), (2,3), (3,5), (4,5), (4,6), (ii) (l, 2), (2,4), (3,4), (3,5), (5,6), (iii) (1, 2), (2, 5), (3, 5), (3,4), (4,6), (iv) (1,2), (2,5), (4,5), (4,6), (v) (1,3), (2,3), (2,4), (4,5), (5,6), (vi) (1,3), (2,3), (2,4), (4,6), (vii) (1,3), (2,3), (2,5), (4,5), (4,6), (viii) (1,3), (2,3), (2,5), (5,6), (ix) (l, 3), (3,4), (2,4), (2,5), (5,6), (x) (1,3), (3,5), (2,5), (2,4), (4,6), (xi) (1,3), (3,5), (4,5), (4,6). 12.5 There are seven such meshes: (i) (2,3), (3,4), (2,4), (ii) (2, 3), (3,4), (4,5), (2, 5), (iii) (2, 3), (3, 5), (2,5), (iv) (2, 3), (3, 5), (4, 5), (2,4), (v)

~,~,(~~,0,~,(~~,

(vi) (2,4), (4,5), (2,5), (vii) (2,4), (4,5), (3,5), (2,3). 12.6 Spanning tree which is an arborescence: (1,2), (2,3), (2,4), (4,5), (4,6). Spanning tree which is not an arborescence: (1,3), (2,3), (3,4), (3,5), (5,6).

165

Answers to Problems

link (j,k)

12.7

(I,2) (1,3) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (4,6) (5,6)

E= node i

1

1

1

0

0

0

0

0

0

0

0

2

-1

0

1

1

I

0

0

0

0

0

3

0

-I

-I

0

0

I

I

0

0

0

4

0

0

0

-I

0

-I

0

1

I

0

5

0

0

0

0

-I

0

-1

-1

0

1

6

0

0

0

0

0

0

0

0

-I

-I

12.8 With the chains enumerated as in the answer to Problem 12.3, chain

mlm2m3m4mSm6m7mSm9 (1,2)

A = link

0 0 0

(I,3)

0 0 0 0 0 0

(2,3)

1

0 0 0 0 0 0

(2,4)

0 0 0

1 1 0 0 0 0

(2,5)

0 0 0 0 0

(3,4)

I

(3,5)

0 0

(4,5)

1 0 0

(4,6)

0

(5,6)

I

0 0 0

1 0 0 0 0

1 1 0

1 0 0 0 0 0 1 0 0

1 0 0 0

0 0

1 0 0

1 1 0

1

I

0

1 0

1

12.9

EA=

1

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

-1

-1

-1

-1

-I

-1

-1

-1

166

Appendiv D

12.10 g = 12. Node 5 :

f25+f35+f45=

f46+f56=4+8= 12=g.

1+3+4=8=f5,. Node 6:

'5

7 -

1 -1

-

1

0

0

0

0

0

0

0

0

0

0

1

1

1

0

0

0

0

0

4

0

0

1

1

0

0

0

1

0

1

1

0

4

0

0

1

3

0

4 4

- 12

0-1-1 0

0

0-1

0

0

0

0-1

0-1-1

0

0

0

0

0

0-1 0

0-1-1

0

12

0

-

0

8 12.1 1 Possible chain flows corresponding to the chains as listed in the answer to Problem 12.3 are

h, = 4, ' 5 -

h,

=

0,

h, = 3,

1 1 1 1 1 1 0 0 0

7

0 0 0 0 0 0 I

0

1 1 1 0 0 0 0 0 0

4

0 0 0 1 1 0 0 0 0

1

0 0 0 0 0 1 0 0 0

-

I

1

4

1 1 0 0 0 0 1

1 0

3

0 0 1 0 0 0 0 0 1

4

1 0 0 1 0 0 1 0 0

4

0 1 0 0 1 0 0 1 0

8

1 0 1 1 0 1 1 0 1

-0-

0 0 0

4 . 1

4 0 -3

Answers to Problems

167

The chain flows are not unique; another possible set is h, = h2 = h 3 = 0,

h4 = 4,

h s = 0,

h 7 = 0,

h s = 4,

h g = 3.

x=

12.12

{l,2,5},

x

=

{3,4,6},

/(X,X) =/13+/23+/24+/56 = 7+0+4+8 = 19,

/(X, X) = /35 + /4S = 3 + 4 = 7, /(X,X)-/(X,X) = 19-7 = 12 =g. 12.13 (a) 8, for chain m 1 (see answer to Problem 12.3). (b) 7, for chain m 2 • (c) C = 102. 12.14 There are 2,,-2 = 24 = 16 cut-sets: X

(X, x)

u(X, X)

{I} {I,2} {I, 3} {I,4} {I,5} {I,2,3} {I,2,4} {I, 2, 5} {I,3,4} {I, 3, 5} {I,4,5} {I,2,3,4} {I,2,3,5} {I,2,4,5} {I, 3,4, 5} {I,2, 3,4, 5}

{(I, 2), (I,3)} {(I, 3), (2,3), (2,4), (2,5)} {(I, 2), (3,4), (3, 5)} {(I, 2), (1,3), (4,5), (4,6)} {(I,2), (1,3), (5,6)} ((2,4), (2,5), (3,4), (3,5)} {(I, 3), (2,3), (2,5), (4,5), (4,6)} {(I, 3), (2,3), (2,4), (5,6)} {(I,2), (3,5), (4,5), (4,6)} {(I, 2), (3,4), (5,6)} {(I,2), (1,3), (4,6), (5,6)} {(2,5), (3,5), (4,6), (4,5)} {(2,4), (3,4), (5,6)} {(I, 3), (2,3), (4,6), (5,6)} {(I, 2), (4,6), (5,6)} {(4,6), (5,6)}

14 15 14 26 23 13 23 23 21 20 31 16 18 27 23 17

12.15 Feasible: 0 12.16

~fij

~ uij

and Ef= g. Not maximal: g* = 13.

(a) The maximal flow is 50 units obtained, for example, by a flow of 50 units on Kearny and Pacific Streets. The cut-set (X, X) = {(47, 52)} for X = {52}, has cut-capacity of 50 units.

168

Appendix D

(b) The maximal flow is 40 units obtained as follows: nodes 1,7,21, 33, 34, 35,28,29 30, 36, 37, 38,47, 52 1, 7, 21, 33, 34, 35, 28, 23 24,29, 30, 36,37,38,47, 52 1 , 7, 8, 9, 23, 17, 16, 24 25, 26, 38,47, 52 I , 7 , 8 , 9 , 10, 16, 11, 19 26, 38, 47, 52

chain flow 10 10 10

10

The cut-set

( X , X ) = {(16,1 I), (16,24), (23,34), (28,39)) for

X

=

{4,5,ll,12,18,19,24,25,26,27,29,30,31,36,

37,38,39,40,41,44,45,46,47,48,50,51,52} has cut capacity of 40 units. Chapter 111 18.1 Figure D.l shows the steps in the labeling procedure and a cheapest route tree (not unique). If the link costs are increased by 2 units, I , 2,4,6 and I , 3,4,6 are cheapest routes but 1,2,3,4,6 is not.

18.2 The cheapest path is 1,2,3,4,6 of cost 7 units. Paths I , 2,3,4,5,6; 1,2,4,6; and 1,3,4,6 are second-to-cheapest paths of cost 8 units. A general algorithm is described in Hoffman, W., and Pavley, R. A Method for the Solution of the Nth Best Path Problem, J . Assoc. Comp. Mach., 6, pp. 506-514 (1959). 18.3 The statement is false. Paths I , 2,4,6,5,3 and 1,2,5,3 are dearest paths of costs 14 units, but I , 2 is not a dearest path (in fact 1,3,5,2 and 1,3,5,6,4,2 are dearest paths of costs 17 units).

169

Answers to Problems

I (4,7)"

6

(0)

Ib)

Figure D.I. Application of the tree-building algorithm. (a) Successive labels, (b) Cheapest tree.

18.4 This matrix algorithm computes at the same time the costs of cheapest routes between all points of nodes. It is stated as a 9-line ALGOL program in the following reference: Floyd, R. W., Algorithm 97, Shortest Path, Comm. ACM, 5, p. 345 (1962). The successive matrices, which are symmetric, are

C(O)

=

0

4

1 0

2 6 8

00

4 2 0 3 5

00

00 00 00

00

6 3 0

00

8 5

00 00 00

I

C(l)

=

0

I

4

I

0

2 6 8

4 2 0

00

3

5

00

00

6 3 0

I

1

8 5

I

0

1

I

1 0

1 0

1

00

I

0

00 00 00

1

00 00 00

170

Appendix D

0137900

0136800

1026800

1025700

3203500 7 6 3 0

e(3)

3203500

=

6 5 301

1 1

875 101 000000 110

98501 000000 10

o

1

1 367 7

102 566 e(4)

=

e(5)

=

e(6)

3 203 4 4

=

6 5 3 0

1 1

764 1 0 1 764 1 1 0 18.5 This matrix algorithm computes at the same time the costs of cheapest routes between all pairs of nodes. The algorithm is described in the following reference: Dantzig, G. B., All Shortest Routes in a Graph, Technical Report No. 66-3, Opns. Res. House, Stanford University (1966). The successive matrices are e(l)

=

[0],

e(2)

=

1 [ 01 0

o e(3)

=

r~ l3

0

=

2

3 6

025 320 3 6 5 3 0

o

367

o =

e(4)

2 0

o e(5)

~],

J,

o

5 6

3 2 0 3 4 6 5 3 0

76410

1 3 677

e(6)

=

2 566

320 3 4 4

6 5 301

76410 I 764 1 1 0

171

Answers to Problems

18.6 The statement is false. It is possible for the cheapest path to go from n, to ni and then via a path from ni to nj whose cost is cheaper than the cost of the link (ni,nj). 18.7 The trace of the cheapest route tree is Link 1

I

2 3 4 5 6 7

Predecessor

Cost

4 1 5 3 6 0 1

245 265 155 180

'P

' C

90

0 280

18.8 The algorithm gives 1,4 as the cheapest route with cost 6, instead of the route 1,2,3,4 with cost 5 units. The algorithm permanently labels node 3 first, which prevents consideration of the alternative route to 3 oia 2. Although 1,2,3,4 is the cheapest route from 1 to 4, it is not true that 1,2,3 is the cheapest route from 1 to 3. 18.9 (i) For each link 0
C C ~ =A 5 + 4 +

4 + 18

+ 9 + 6 = 46.

(ii) With the same notation as in the Table 15.2 (Chap. HI), we can tabulate the chains, chain flow, route costs, and the sums over dual variables as follows: Chain

CJ

hJ

8 7

0

9 9

0

8 10 9 8 10

-

2

0 3 0

0 1 0

172

Appendix D

(iii) If v*-Cpi* 0, then fi* = ui, verified for i = 3; Ifhj*>O, then v*-Xpi*=Cj(O,D); verifiedforj=2,5,8; Iff;.* < ui, then pi* = 0, verified for all i # 3. (iv) All chains are available for flow from chain m2 and all have chain costs 2 C2= 7. All chains except m,,m 2 ,m3are available for flow from chains m, and ma and have chain costs 2 C, = Ca = 8. 18.10 (i) C = x c i f i = 9 1 . (ii) Chain

Cpi

v-xpi

C,

8 7 9 9 8 9 9 8

8

10

hJ

7

9 9 8 10

9 8 10

Also Y = ~ g - C ~ * ~ i = ( 1 0 ) ( 1 1 ) - 6 - 5 - 8=91. (iii) If v*-Cpi* < C,(O,D), then hi* =0, vertified f o r j = 6; If pi* > 0, then fi* = ui,verified for i = 1,6,9;

If hj* > 0, then v* - 1 pi* = C j ( 0 ,D), verified for j = 2,5,7,8,9; If fi* < ui, then pi* =0, verified for i =2,5,7,8,10. (iv) All chains except m4 and m 5 are available for flow from chain m, and have chain costs > C2= 7. Chains m4,m6,and m, are available for flow from m5,chain costs 2 C, = 8. Chain m, is available for flow from m7,chain cost 2 C7= 9. Chains m, and m, are available for flow from ma, chain costs 2 C, = 8. No chain is available for flow from m,.

173

Answers to Problems

18.1 1 (i) The chains with route costs are Chain

0-D pair

k

j

Oi-Di

1

01-D2

2

1 2 3 1

,(k) J

1,6,8,10 2,4,6,8,10 2,5,7,9,10 1,6,8,11 2,4,6,8.11 2,5,7,9,11 3,4,6,8,10 3,5,7,9,10 3.4,6,8,11 3,5,7,9,11

2

Oz-Di

3

02-D2

4

Route cost

3 1 2 1 2

Chain flow

C:L)

hp)

7 6 8 9 8 10 10 12 12 14

0 35 0 0 20 0 15 0 30 0

(ii) The all-or-nothing cheapest route assignment gives the chain flows hf) listed above with consequent link flows: Link Commodityflow

Link flow

i

1

2

3

fi”)

0

35

fi”’

0 0 0

20 0 0

0

55

A“’ A(”’ A

4

5

6

7

35 20 15 30 30 45 100

0 0 0 0 0

35 20 15 30 100

0 0 15

8

9

1

35 20 15 0 30 0 100

0 0 0

0

1 35 0 15 0 50

0 0

0 0 0

1 0 20 0 30 50

Network cost is C = 880 units. 18.12 Origin

Link copy flow Link flow

Copy

0 1

1

0 2

2 i

A’

fr2 fi

1

2

0 0

0

Links in tree 2,4,6,8,10,11 3,4,6.8,10,11 3

55 0 55

Copy flow

4 0 45 45

5

55 45 100

6

u1 = 55

u2 = 45

7

0 55 0 45 0 100

8

9

0 55 0 45 0 100

1

0 0 0 0

1 35 15 50

1 20 30 50

174

Appendix D

18.13 Copy

Destination

... -

_---~---

I 2

D, D2

Copy flow

Links in tree

- - - - _ ..

-----

2,3,4,6,8,10 2,3,4, 6,8, II 2

Link

3

4

v' v2

= 50 = 50

7

8

6

5

9

10

II

0 0 0

50 0 50

0 50 50

-----_.-,------,------ - - - _ .

f?

I,

Link flow

35 20 55

0 0 0

f,'

Copy flow

15 50 30 50 45 100

0 50 0 50 0 100

0 50 0 50 0 100

18.14 With the chains enumerated as in the answer to Problem 18.11, the chain flows for the two loadings (i), (ii) are Q-D pair

k

j

-----

-----

I 2 3 I 2 3 I 2 I 2

O,-D,

O.-D 2

2

02- D ,

3

02- D2

4

WI

Chain flows (i)

(ii)

--._.-

0 35 0 5 15 0 0 15 0 30

35 0 0 15 5 0 15 0 30 0

The resulting link flows are 2

Link ..

--_.~--------_

3

(i) Link flow (ii) Link flow

5 50

fi fi

4

5

- ----_._.-

-------

50 5

45 45

50 50

6

8

7

9

45 55 0 100

45 55 0 100

10

II

-----

-------.'-.

45 0

50 50

50 50

Network costs are (i) C = 975, (ii) C = 930. Both these network flows satisfy the first extremal principle. 18.15 In the notation of(72)-(75), Sect. 16, Chap. Ill, and the answer to Problem 18.11, the problem of minimizing the network cost can be expressed as the LP h\ I), h~l),

h~I),

h\2), h~2),

h~2),

h\3), h~3), h\l)

h\4), h~4)

+ h~l) + h~1)

h\2) + h~2)

+ h~2)

~

0,

= 35, = 20,

175

The quantity s is a nonnegative slack flow. By means of any of the standard LP algorithms, this problem can be expressed as hi” = 35 - h(’) - h(’) 2 3 ,

- j $ 2 ) - h(3) - hi4) + s, 2

hi2) = 15 +

+

hi2’ = 5 h p + h$2’ + hi41 - s, j i 3 ) = 15 - h(3) 2 7

h(4) = 30 - h(4) 1 2 9 C = 930

+ hS1)+ hi2’ + hi3)+ hi4) + s.

The optimal solution which minimizes Cis read off as hi’) = 35, h\2) = 15, hS2) = 5, hi3)= 15, hi4) = 30 with all other chain flows zero, and C* = 930. This is the solution for Case (ii) in the previous problem. 18.16 With the same notation as in the answer to Problem 18.1 1, the nonzero chain flows for the user-optimized traffic pattern are : my)

cp

hp)

O1-DI Ol-Dl OI-D2 01-D2 Oz-Dl

ml’) m:’) mi2) my)

14 16

14

02-DZ

mi4)

h 35-h 40-h h- 20 15 30

16

mi3)

16 18

The flow on chain mi1) can be taken equal to any value h, 0 < h < 20. The optimal chain flow pattern is therefore not unique but the unique optimal link flow pattern is: fi*

f,*

=

= 40, f2* = 15,

j3*

40, f7* = 60, fs* = 40,

= 45, fg*

f4*

=

0,

= 60, f:o

f5*

=

= 50,

60, f:l

= 50.

176

Appendix D

18.17 Using the same notation as in the answer to Problem 18.11 the nonzero chain flows for the system-optimized traffic pattern are: O1-D1 OI-DI 01-D2 O1-Dz 02-D1 02-DZ

my)

Cp

hy)

m',l)

10 15 12 17 17 19

h 35-h 30-h h- 10 15 30

m'"

mi') miz) mi3) mi4)

The flow on chain mi1)can be taken equal to any value h, 0 < h < 10. The optimal chain flow pattern is therefore not unique but the unique optimal link flow pattern is: fl* f6

*- 30,

= 30, fi* = 25, f3* = 45, f7* =

70, f s * = 30,

f4*

= 0,

= 70, f:o

fg*

f5*

= 70,

= 50, f:1

= 50.

18.18 Using Appendix B we can write the maximal flow problem as the following primal linear program, with its dual obtained by using dual variables - li for i = 1,2, ..., n and p i j for ( i , j )E L:

Dual D

Primal P f.1

20

fij

20

fij

A,,--A1

pij

< uij

Cfij-f.1 A(1)

+ li - A j 2 0

( i , j )E L

pij 2 0

(i,j ) E L A1 unrestricted in sign

= O

Ai unrestricted in sign i = 2, ...,n - 1

fij - B(i) 1hi = 0

A(i)

A,, , unrestricted I& =0 , in sign -C u i j p i j = u(max). = z(min) -

-fnl

2 1

B(n)

(i.j)EL

For any cut-set ( X , 1)separating the origin from the destination,

and Pij

=

1

if(i,j) E (X,X)

0

otherwise

177

Answers to problems

give a feasible solution of the dual with = -u(X,X).

0

By (6) in Appendix B, z = -S,1 2 v = - u ( X , X )

or, sinceS,, = g = flow value,

< U(X,X)

g

as given in (lo), Sect. 9, Chap. 11. For optimal primal and dual solutions,

:f

or

=

C u i j p*i j ,

g* = u ( X * , X * )

which proves the max-flow min-cut theorem. The complementary slackness relations are : if I,* - 11* > 1

then : f = 0;

if pc

then

+ Ii* - Ij* > 0

if pi:. > 0

then

fz

=0

( i , j ) E L;

= uij

( i , j ) E L;

if f:

>0

if

>0

then pi";.+ Ii* - I j *

if

< uij

then pi:. = 0.

then I,* - 11*= 1,

=

0,

The dependence of the optimal dual variables on the optimal link flows can be expressed as follows:

11*- I,* and A,*

-A,*

< -1 =

-1

ILj*- Ii*

<0

lj*- Ai*

=

0

ij*- Ii* 2 0 p?. IJ = 0

* pij

20

Compare Fig. 15.1, Chap. 111.

for

l:f

for : f

= 0, = 0;

for f: = 0, for

0
for f; = uij ; for

< uij,

for f$ = u i j .

178

Appendix D

Chapter IV

27.1

From (13), Sect. 21,

, 2(5-5)+6(5-3) 12 c 13 = 2+6 = g' , 2(9-5)+6(9-3) c2 3 = 2+6 ,

c 3 3--

44

=g'

2(7-5)+6(7-3) 28 2+6 -g.

The new table of costs with the new values

a/ and b/

is

a.' I

5

4

1.5

5

10 8

5.5

5

9 3.5

5

9

b/

6

8

By (18), Sect. 21,

P3

'_ -

2(-5)+6(-3) _ 2+6 -

28

-g'

giving a compressed trip table with new implicit prices a.. '

•

3

o o P/

5

3

2

9

0

5

7

0 -I -3.5

The optimality of these flows is verified from the complementary slackness relations:

Answers to Problems

179

f;2

= 3,

f;3

= 2,

fj3

=

+ =8 = 42, a2’ + 83’ = 5.5 = 43, + = 3.5 = , ~ 2 ‘ B2’

5,

4 3

83’

a2’

+ B1’ = 9 < C i l = 10, fil = 0, a3‘ + = 7 < Cil = 9, fj, = 0, a2‘

B1’

~ 3f ’ B2’

=

6<4

2

=

9,

fj2

= 0.

It is interesting to note that the trip table

with mn - (m + n - 1) = 4 zero entries, is another optimal solution for the new Hitchcock problem, but it is not obtained by compressing the original table.

27.2 From (12), Sect. 21, c;, =

Ix5+6x4 29 =1+6 7’

c;l =

Ix9+6x9 = 9, 1+6

and from (13), Sect. 21,

1(9+0)+6(91+6 so that the new cost table is Cil =

I)

Ui‘

29

3 215

7

9

8 415

-.__

6,‘

7 2 6

57 7’

=-

180

Appendix D

By (18), Sect. 21,

pi = 1

1(0)+6(-1) 1+6

=_i

7"

The compressed trip table with new implicit prices is

4

0

1

5

3

2

0

9

o

0

5

7

6

-- -5 -3 7 The complementary slackness relations are: 01: 1

+ p'1

121 =

3,

01:2

1;2 =

2,

0I:z' + P2'

1~3

01:3

I

=

57 7 =

, cu,

I

C2l'

C22'

= 4 =

= 5,

, + P'1 = 43 7 <

01: 3'

27.3

, + P'1 = 7 29 =

+ P2'

= 2

,

C3 l

< C32

= 9, = 8.

1~1

= 0,

1~2

= O.

From (27), Sect. 22, the trip table is calculated to be ai

bj

4

8

12

16

40

3

6

9

12

30

2

4

6

8

20

2

3

4

10

10 20 30 40

100 = v

181

Answers to Problems

Combining centroids 2 and 3 giv.es

[ bi

a;

4 20 16 2:

40 50

2:]

10

10 50 40

100 = U’

while removing centroid 2 gives a,’

bj’

7 21 28

56 = U’

27.4 The following results were obtained by computer calculations:

I

0.37 2.20 0.55

fij =

1.88

0.30 2.17 1.04 1.49 0.33 1.63 0.41 2.63

The values x1 = 0.616,

x2 = 1.419,

x3 = 1.312,

y1 = 0.115,

y, = 0.554,

y, = 0.113,

1

.

y4 = 0.310,

y = 0.211,

were obtained at the end of the iterations. 27.5 The design year productions and attractions are

Design year Zone i

Production ai

Attraction bi

1 2 3

14 33 28

33 28 14

182

Appendix D

Consider the iterative scheme

for k = 1,2, .... Applying these to the given data yields the successive iterates :

37 24

14

34 27 14

33 28

14

Calculations have been approximate and answers rounded off to give integers. This simple growth factor and scaling technique was developed by Fratar as one of the earliest trip distribution models. This and the next two problems, together with the answers, have been taken from the Manual for the Urban Planning System/360 Trip Distribution Programs, Bureau of Public Roads, U S Department of Transportation, 1968.

183

Answers to Problems

27.6 The successive values of bj are

i

1

by) bJ2) b$3)

33 28 27

2

3

28 33 34

14 14 13

with corresponding trip tables

[Ay)] =

[fh3)] =

[

2

10

2

14

19

8

6

33

17

5

6 1 28

38 23

14

34 27

14

[ ,:

17 10 l:

33 28

1I f 6

33.

14

Calculations have been approximate and answers rounded off to give integers. 27.7 The successive values of bj are

i b$l) b$z) b$3)

1

2

3

33 35 31

28 41 48

14 13 14

184

with corresponding trip tables

[ ’: :]

13

[fl))]

=

1;

25,

12

3

12

27

31

19

15

65

31 24

13

68

[ ‘i :]

13

1

[fi’i)] =

20

12

29.

3

12

27

30 24

15

69

Calculations have been approximate and answers rounded off to give integers. 27.8 The iterative steps and the destination-optimal distribution are

obtained as follows:

1

2

3

4

5

6

7

7 6 6 6

2 2 2 2

1 1 1 1

7 7 7 7

5 5 5 5

2 2 1 3

4 4 4 4

27.9 (a) We first find stationary points of

185

Answers to pmblems

The partial derivatives of Y with respect to pij are

a 2- - - 1 - lnpij - lli - p j - v -

i,j # 1,1,

aPij

= - 1 - 1 n p i j - A i - p . - 6J - v

i , j = 1,l.

The optimal flows can, therefore, be rewritten in the form

~,*= i XiYj

i , j # 1,1, i , j = 1,1,

= $xl y1

where 4 = e-' and row and column sums are

= xl($y1+y2+ ...+y,)

= ($x,+x2+

...+xJy1

= ui,

i# 1

= ul,

i = 1,

- uj,

j # 1

= Ul,

j = 1.

The ratio of row totals is i # 1.

ui/u2 = xi/x2, Similarly,

ujlu2 = ~

j l3 ~ 2j

# 1-

The optimal solution for p$ can be written in terms of these ratios and the given value of p1 = k. We obtain

P:*

+ P:2-

1- v 1 02

= u2.

186

Appendix D

Thus,

i,j ¥- 1,2. The computation proceeds by finding pT2, pit> pi2, PtJ, other p~. (b) For Ptt = 0.01, the optimal solution is Ui

Vj

0.01 0.08 0.16

0.25

0.13 0.04 0.08

0.25

0.26 0.08 0.16

0.50

0.40 0.20 0.40

pij and then all

AUTHOR INDEX

Numbers in parentheses are reference numbers and indicate that an author’s work is referred to, although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed. B Bay Area Transportation Study Commission, 5(1), 13(1), 14 Beckmann, M. J., 17(6), 45,108, 145 Bellman, R.,51,103 Berge, C.,17(2), 44 Blackburn,J. B., 109 Blunden, W. R.,144(15), 148 Brokke, G . E., 64(8), 103 Burrell, J. E.,65,105,142(17), 149 Busacker, R.G . 17(3), 44

Dijkstra, E. W., 52(6), I03 Dreyfus, S.E., 51, I02

F

Fairthorne, D. B., 144(14), 148 Floyd, R.W.,169 Ford, L.R., 17(1),43(1), 44,75(1), 107, 119(1) Fulkerson, D.R.,17(1),43(1),44,75(1), 107,119(1) Funk, M.L.,109

G

C

Caldwell, T., 57,61,62,103 Charnes, A., 100,108,149 Control Data Corporation, 104 Cooper, W. W., 100,108,149

D Dafermos, S. C., 107 Dantzig, G . B., 66(18), 67(18), 106, 107, 119(18), 170 Dearinger, J. A., 144(13), 148

Gale, D., 138 (ll), 147 Ghouila-Houri,A., 17(2), 44 Gibert, A., 101 (27),109 Golob, T. F.,145 Greater LondonCouncil, 10(2), 13(2), 14

H Harary, F., 17(5), 44 Heanue, K.E., 134(12), 144(12), 147 Heggie, I. G., 136(8), 144(8), 146,147

187

188

Author Index

Highway Research Board, 41,45 Hitchcock, F. L.,116,118-121,144 Hoffman, W.,168

I Irwin, N. A., 142(16), 148 J Jansen, G. R., 63,I05 Jorgenson, N. O., 66(17), 67(17), 70, 96(17), 106

K

Kaufmann, A., 17(4), 44 Kirby, R. F., 51(4), 57(4), 63(11), 103, 104, 138,141(10), I47 Kitchen, J. W.,108,160(23)

L

Lawson, M. C., 144(13), 148 M McDonald, W. R., 144(15), 148 McGuire, C. B.,17(6), 45 Metropolitan Corporation of Greater Winnipeg, 13(3), 14 Meyers, D. A., 142(17), I49 Michaels, R. M., 65,105 Murchland, J. D., 51, 102, 122,144 N

Nash, J., 107

P Pavley, R.,168 Pinnell, C., 101,109 Potts, R.B., 51 (4),57(4), 103

Powell, T J., 142(17), 149 Pyers, C. E.,134(12), 144(12), 147 S Saaty, T.L., 17(3), 44 Sasaki, T., 131, 133(4), 145 Satterly, G.T., 101,109 Sema, Group Metra France, 62(10), 104 Shapley, L.S.,138(1 I), 147 Sinkhorn, R., 133, 135(6), 145 Smith, W. S., 13(4), 14 Snell, R.R., 109 Sparrow, F. T.,I07 Stouffer, S. A., 136,147 Stover, V. G.,64(14), 105

T Tanner, J. C.,146 Tomlin, J. A., 101, 109, 131, 133(5), 143(18), 145,149 Tomlin, S. G., 131, 133(5), 145 Traffic Research Corporation, 142 Tresidder, J. O.,142(17), I49

U

U.S. Department of Transportation, 182

V

Veinott, A. F., 106 Von Cube, A. G., 142(16), 148 W Wachs, M.,63(12), 104 Wardrop, J. G., 49,102 White, D.J., 145 Wilson,A.G., 122(3), 144,146,147 Winsten, C. B.,17(6), 45

SUBJECT INDEX

Numbers in italics indicate the first occurrence or formal definition of a term. A Accessible node, 22 Admissible route, 61 After nodes, 27 Algorithms gravity model, 135-136 intervening opportunities, 138 matrix scaling, 145-146 mean trip length model, 132-133 out-of-kilter, 66, 75-86 preferencing model, 140-141 shortest path, 102-104 tree-building, 52-56, 60 All-or-nothing assignment, 64 A-node, I8 Arborescence, 26 Assignment all-or-nothing, 64 associated traffic, 95-96 cheapest route, 63-64 combined distribution, 116, 141-143 congested, 100-101 diversion, 64 future traffic, 13

multicommodity distribution-assignment, 143 multiple route, 65 traffic, 12, 115-116 Attraction nodes, 27-29 Available chains, 70 Average costs of chain flow, 95, 159-162

B Base year inventory, 11-12 Bipartite graph, 18 B-node. 18

C

Capacitated network, 4 1 4 3 Centroid, 4,26-29, 1 15-1 16 Chain, 20,21-22 available, 70 flow, 71 traffic pattern, 33 unavailable, 70 Cheapest route assignment, 63-64 City street network, 2, 3, 40 189

190

Subject Index

Coding a network, 5 Commodity, 34, 96 Complementary slackness, 67, 76, 158, 177-178 Complete graph, 18 nonplanar, 9 Compressed network, 36-37 Compressibility, 36, 120-121, 128-130, 136 Congested assignment, 100-101 Connected directed graph, 22 Connected nodes, 22 Conservation equations, 27, 30-31, 108 Conservation principle, 26-38 COPY, 34 Cordon line, 6 Costs, 38 link, 38 flow independent, 39 route, 39 Cut capacity, 42, 43 Cut-set, 23, 24 Cycle, 19, 21

D Dearest path, 110 Destination node, 20, 29 Destination-optimal, I39 Deterrence function, 134-135, 141-142 Directed graph, 18 Directed link, 2 Distribution combined with assignment, 116, 141I43 compressibility, 117, 120-121, 128130, 136 destination-optimal, 139 entropy, 116, 123-128 equilibrium, 122 formulation, I 16- I I 8 mean trip length, 130 gravity, 116, 133-136 origin-optimal, 139 preferencing, 116, 138-141 proportional, 123-128 trip, 12, 115-1 I6 Districts, 4 Diversion assignment, 64

Dual linear program, 67, 119, 121, 157158, 176-177

E

Economic analysis, 13 Enroute points, 26 Entropy models, 121-136 Expanded network, 36, 38 Extremal principles, 50, 68

F Flow independent link cost, 39 Flow value, 30, 32 Flow-augmenting path, 78 Flows chain, 32-36, 70-75 commodity, 34 copy, 34, 35 link, 6 6 7 1 network, 42 Forward link, 21 Future traffic analysis, 13 G Graphs, 8 accessible, 22, 23 bipartite graph, 18 complete, 18 connected directed, 22 mixed, 24 partial, 18 subgraph, 18 undirected, 24 Gravity model, 122, 133-136 H Hitchcock model, 118-121 Home node, 25, I 10 I Interactance model, 134 Intermediate node, 4, 27 Intervening opportunities model, 137I38 Inventory of main roads and transit services, 12 of planning factors, 12 of travel patterns, 12

Subject Index

Joined nodes, 18

191 J

K

Kilter numbers, 76, 77 Kirchhoffs law, 26-27,29,36, 38

L

Labels, 55 Lagrange function, 123, 134 Lagrange multiplier, 123-124, 134 Linear programming complementary slackness, 67, 76, 158, 177-1 78 dual, 67, 119, 121, 157-158, 176-177 primal, 67, 157-158, 176- 177 Link, 2, 18 capacity, 41 dummy, 4-5 kilter numbers, 77 saturated, 70 unsaturated, 70, 72 in-kilter, 76-77 out-of-kilter, 76-77 Link-chain incidence matrix, 33 Link cost, 38 Link flow traffic pattern, 32 Loop, 18 LTS program, 142- I43

M Main road network, 4, 5-7 Marginal costs of chain flow, 91, 95, 159-162 Matrix distribution, I 1 5 link-chain incidence, 33, 35, 45 node-link incidence, 31, 34, 45 scaling theorem, 145 Mean trip length, 130 model, 130-133 Mesh, 21,22 Minimum network cost, 65-75 Mixed graph, 24 Models, 115-144 entropy, 121-136 gravity, 122, 133-136 Hitchcock, 118-121

interactance, 134 intervening opportunities, 137-1 38 mean trip length, 130-133 opportunity, 136-141 preferencing, 138-141 proportional, 123-130 Multicommodity distribution assignment, 143 Multiple 0-D network, 34-36 Multiple route assignment, 65 N Nash equilibria, 107 Network capacitated, 41-43 city street, 2, 3, 40 complete nonplanar, 9 compressed, 36-37 Cost, 40, 65-101 entropy, 121-136, 122 evaluation, 13 expanded, 36, 38 feasible flow, 42 flow value, 30, 32 main road, 4, 5-7 minimum cost, 65-75 multiple 0-D, 34-36 with prohibited turn, 60 pseudo-, 57, 61, 62 single 0-D, 29, 30-34 spider web, 9 traffic desire, 8-9 transportation, I , 2 with turn penalties, 56-62 Node-link incidence matrix, 31 Nodes, 18, 2 accessible, 22 attraction, 27-29 centroid, 26-29 connected, 22 destination, 20, 29 distinct, 60 home, 25, 1 10 intermediate, 4, 27 joined, 18 labelled, 55 origin, 20, 29 predecessor, 154-155

192

Subject Index

production, 27-29 sink,27 source, 27

o

Opportunity models, 136-141 Origin node, 20, 29 Origin-optimal, 139 Out-of-kilter, 75, 76-77 p Partial graph, 18 Path, 21 Planning factors, 12 Preferencing model, 138-141 Primal linear program, 67,157-158,176177

Principles available chains, 50, 75 compressibility, 36 conservation, 26-38 equilibrium distribution, 122 extremal, 50 Kirchhoff's law, 26-27, 29, 36, 38 Maximum entropy, 122, 123 minimum network cost, 91-95 separability, 36 system-optimized traffic patterns, 50 user-optimized traffic patterns, 50, 89 Wardrop's, 49-50 Production nodes, 27-29 Proportional model, 123-130 Pseudo-network, 57, 61-62 R

Reverse link, 21 Route, 17 admissible, 61 cost, 39-40, 70 S Saturated link, 70 Screen line, 6 Sectors, 4 Separability, 36, 38 SHARE program, 106 Single O-D network, 29,30--34 Sink,27 Skim tree, 55, 116

Source, 27 Spanning tree, 24 Spider web network 9 Subgraph, 18 ' System-optimized traffic pattern, 50 T Theorems cheapest route, 51, 153 matrix scaling, 145-146 max-flow, min-cut, 43 net flow, 30 system-optimized traffic patterns 9195 ' turn penalty, 59 user-optimized traffic patterns 90 Total link cost, 38 ' Traffic desire 9, 115 network, 8-9 Transportation networks, 1-10,2, 13 planning, 10-13, 115-117 studies, 5, 6, 13-15, 31, 137, 142, 148 Travel forecasts, 12 Travel patterns, 12 TRC program, 142 Tree, 24 -building algorithms, 52-56, 60, 169171

skim, 55, 116 spanning, 24 trace, 55-56, 171 Trip distribution, 9, 12, 115-152 Trip generation, 12 Trip length frequency distributions, 130 Turn penalties, 56 U

Unavailable chains, 70 Undirected graph, 24 Undirected link, 2, 24 Unsaturated link, 70, 72 User-optimized traffic pattern, 50, 89 Zones, 4 destination, 115 origin, 115

z